(*** helper for §35 Step II: real-valued extension with range in [-1,1] ***)
(*** LATEX VERSION: Step 2 (core): extend f:A→(-1,1) by an infinite series of Urysohn functions; here packaged as existence of a continuous gR:X→R agreeing with f on A and mapping X into [-1,1]. ***)
(*** helper for §35 Step II: nonempty closed subset case, real-valued extension ***)
(*** LATEX VERSION: Step II (nonempty A): construct a real-valued continuous extension gR:X->R agreeing with f on A and bounded in [-1,1]. ***)
L174349
Let X, Tx, A and f be given.
L174350
Assume Hnorm: normal_space X Tx.
L174351
Assume HA: closed_in X Tx A.
L174352
Assume HAnemp: A ≠ Empty.
L174353
Assume Hf: continuous_map A (subspace_topology X Tx A) (closed_interval (minus_SNo 1) 1) (closed_interval_topology (minus_SNo 1) 1) f.
L174355
We will prove ∃gR : set, continuous_map X Tx R R_standard_topology gR ∧ (∀x : set, x ∈ A → apply_fun gR x = apply_fun f x) ∧ (∀x : set, x ∈ X → apply_fun gR x ∈ closed_interval (minus_SNo 1) 1).
L174359
Set I to be the term closed_interval (minus_SNo 1) 1.
L174361
We prove the intermediate claim HTx: topology_on X Tx.
(*** TeX Step II series construction: build gR as a uniformly convergent series of Urysohn-step functions (formalized below). ***)
L174363
An exact proof term for the current goal is (normal_space_topology_on X Tx Hnorm).
L174363
We prove the intermediate claim HAsubX: A ⊆ X.
L174365
An exact proof term for the current goal is (closed_in_subset X Tx A HA).
L174365
We prove the intermediate claim Hf_fun: function_on f A I.
L174367
An exact proof term for the current goal is (continuous_map_function_on A (subspace_topology X Tx A) I (closed_interval_topology (minus_SNo 1) 1) f Hf).
L174368
We prove the intermediate claim Hf_R: function_on f A R.
L174370
Let x be given.
L174370
Assume HxA: x ∈ A.
L174370
We prove the intermediate claim HfxI: apply_fun f x ∈ I.
L174372
An exact proof term for the current goal is (Hf_fun x HxA).
L174372
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun f x) HfxI).
L174373
Set I0 to be the term closed_interval (minus_SNo one_third) one_third.
L174376
Set T0 to be the term closed_interval_topology (minus_SNo one_third) one_third.
(*** TeX Step II outline: obtain g0 by Step I, then iterate on residuals and take a uniformly convergent series. ***)
L174377
We prove the intermediate claim Hexg0: ∃g0 : set, continuous_map X Tx I0 T0 g0 ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval one_third 1) ∩ I) → apply_fun g0 x = one_third).
L174385
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A f Hnorm HA Hf).
L174385
Apply Hexg0 to the current goal.
L174386
Let g0 be given.
L174387
Assume Hg0.
L174387
We prove the intermediate claim Hg0contI0: continuous_map X Tx I0 T0 g0.
L174389
We prove the intermediate claim Hleft: continuous_map X Tx I0 T0 g0 ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third).
L174393
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 g0 ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A f ((closed_interval one_third 1) ∩ I) → apply_fun g0 x = one_third) Hg0).
L174399
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 g0) (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third) Hleft).
L174404
We prove the intermediate claim Hg0contR: continuous_map X Tx R R_standard_topology g0.
L174406
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L174407
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology g0 Hg0contI0 (closed_interval_sub_R (minus_SNo one_third) one_third) R_standard_topology_is_topology_local HT0eq).
L174412
Set f1 to be the term graph A (λx : set ⇒ add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))).
L174414
We prove the intermediate claim Hfung0: function_on g0 X I0.
(*** Residual on A: f1 = f - g0|A. ***)
L174416
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 g0 Hg0contI0).
L174416
We prove the intermediate claim Hf1_total: total_function_on f1 A R.
L174418
Apply (total_function_on_graph A R (λx : set ⇒ add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x)))) to the current goal.
L174418
Let x be given.
L174419
Assume HxA: x ∈ A.
L174419
We will prove add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x)) ∈ R.
L174420
We prove the intermediate claim HfxR: apply_fun f x ∈ R.
L174422
An exact proof term for the current goal is (Hf_R x HxA).
L174422
We prove the intermediate claim HgxI0: apply_fun g0 x ∈ I0.
L174424
An exact proof term for the current goal is (Hfung0 x (HAsubX x HxA)).
L174424
We prove the intermediate claim HgxR: apply_fun g0 x ∈ R.
L174426
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun g0 x) HgxI0).
L174426
We prove the intermediate claim HmgxR: minus_SNo (apply_fun g0 x) ∈ R.
L174428
An exact proof term for the current goal is (real_minus_SNo (apply_fun g0 x) HgxR).
L174428
An exact proof term for the current goal is (real_add_SNo (apply_fun f x) HfxR (minus_SNo (apply_fun g0 x)) HmgxR).
L174429
We prove the intermediate claim Hg0pair: (continuous_map X Tx I0 T0 g0 ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third)) ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval one_third 1) ∩ I) → apply_fun g0 x = one_third).
L174437
An exact proof term for the current goal is Hg0.
L174437
We prove the intermediate claim Hg0_on_B: ∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third.
L174441
We prove the intermediate claim Hleft: continuous_map X Tx I0 T0 g0 ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third).
L174445
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 g0 ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A f ((closed_interval one_third 1) ∩ I) → apply_fun g0 x = one_third) Hg0pair).
L174451
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 g0) (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third) Hleft).
L174456
We prove the intermediate claim Hg0_on_C: ∀x : set, x ∈ preimage_of A f ((closed_interval one_third 1) ∩ I) → apply_fun g0 x = one_third.
L174460
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 g0 ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A f ((closed_interval one_third 1) ∩ I) → apply_fun g0 x = one_third) Hg0pair).
L174466
We prove the intermediate claim Hf1_apply: ∀x : set, x ∈ A → apply_fun f1 x = add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x)).
L174470
Let x be given.
L174470
Assume HxA: x ∈ A.
L174470
rewrite the current goal using (apply_fun_graph A (λx0 : set ⇒ add_SNo (apply_fun f x0) (minus_SNo (apply_fun g0 x0))) x HxA) (from left to right).
Use reflexivity.
L174472
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L174474
Set I3 to be the term closed_interval one_third 1.
L174475
Set I2 to be the term closed_interval (minus_SNo two_thirds) two_thirds.
L174476
Set B to be the term preimage_of A f (I1 ∩ I).
L174477
Set C to be the term preimage_of A f (I3 ∩ I).
L174478
We prove the intermediate claim Hf1_range: ∀x : set, x ∈ A → apply_fun f1 x ∈ I2.
L174481
Let x be given.
L174481
Assume HxA: x ∈ A.
L174481
We prove the intermediate claim HfxI: apply_fun f x ∈ I.
L174483
An exact proof term for the current goal is (Hf_fun x HxA).
L174483
We prove the intermediate claim HxX: x ∈ X.
L174485
An exact proof term for the current goal is (HAsubX x HxA).
L174485
Apply (xm (x ∈ B)) to the current goal.
L174487
Assume HxB: x ∈ B.
L174487
We prove the intermediate claim Hg0eq: apply_fun g0 x = minus_SNo one_third.
L174489
An exact proof term for the current goal is (Hg0_on_B x HxB).
L174489
We prove the intermediate claim Hf1eq: apply_fun f1 x = add_SNo (apply_fun f x) one_third.
L174491
rewrite the current goal using (Hf1_apply x HxA) (from left to right).
L174491
rewrite the current goal using Hg0eq (from left to right) at position 1.
L174492
We prove the intermediate claim H13R: one_third ∈ R.
L174494
An exact proof term for the current goal is one_third_in_R.
L174494
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L174496
rewrite the current goal using Hf1eq (from left to right).
L174497
We prove the intermediate claim HfxI1I: apply_fun f x ∈ I1 ∩ I.
L174499
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun f x0 ∈ I1 ∩ I) x HxB).
L174499
We prove the intermediate claim HfxI1: apply_fun f x ∈ I1.
L174501
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun f x) HfxI1I).
L174501
We prove the intermediate claim H13R: one_third ∈ R.
L174503
An exact proof term for the current goal is one_third_in_R.
L174503
We prove the intermediate claim H23R: two_thirds ∈ R.
L174505
An exact proof term for the current goal is two_thirds_in_R.
L174505
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L174507
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L174507
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L174509
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L174509
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L174511
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L174511
We prove the intermediate claim HfxR: apply_fun f x ∈ R.
L174513
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun f x) HfxI1).
L174513
We prove the intermediate claim Hfx_bounds: Rle (minus_SNo 1) (apply_fun f x) ∧ Rle (apply_fun f x) (minus_SNo one_third).
L174515
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun f x) Hm1R Hm13R HfxI1).
L174516
We prove the intermediate claim Hm1lefx: Rle (minus_SNo 1) (apply_fun f x).
L174518
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun f x)) (Rle (apply_fun f x) (minus_SNo one_third)) Hfx_bounds).
L174519
We prove the intermediate claim Hfxlem13: Rle (apply_fun f x) (minus_SNo one_third).
L174521
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun f x)) (Rle (apply_fun f x) (minus_SNo one_third)) Hfx_bounds).
L174522
We prove the intermediate claim Hf1R: add_SNo (apply_fun f x) one_third ∈ R.
L174524
An exact proof term for the current goal is (real_add_SNo (apply_fun f x) HfxR one_third H13R).
L174524
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun f x) one_third).
L174526
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun f x) one_third Hm1R HfxR H13R Hm1lefx).
L174526
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun f x) one_third).
L174528
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L174528
An exact proof term for the current goal is Hlow_tmp.
L174529
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun f x) one_third) (add_SNo (minus_SNo one_third) one_third).
L174531
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun f x) (minus_SNo one_third) one_third HfxR Hm13R H13R Hfxlem13).
L174531
We prove the intermediate claim H13S: SNo one_third.
L174533
An exact proof term for the current goal is (real_SNo one_third H13R).
L174533
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun f x) one_third) 0.
L174535
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L174535
An exact proof term for the current goal is Hup0_tmp.
L174536
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L174538
An exact proof term for the current goal is Rle_0_two_thirds.
L174538
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun f x) one_third) two_thirds.
L174540
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun f x) one_third) 0 two_thirds Hup0 H0le23).
L174540
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun f x) one_third) Hm23R H23R Hf1R Hlow Hup).
L174543
Assume HxnotB: ¬ (x ∈ B).
L174543
Apply (xm (x ∈ C)) to the current goal.
L174545
Assume HxC: x ∈ C.
L174545
We prove the intermediate claim Hg0eq: apply_fun g0 x = one_third.
L174547
An exact proof term for the current goal is (Hg0_on_C x HxC).
L174547
We prove the intermediate claim Hf1eq: apply_fun f1 x = add_SNo (apply_fun f x) (minus_SNo one_third).
L174549
rewrite the current goal using (Hf1_apply x HxA) (from left to right).
L174549
rewrite the current goal using Hg0eq (from left to right) at position 1.
Use reflexivity.
L174551
rewrite the current goal using Hf1eq (from left to right).
L174552
We prove the intermediate claim HfxI3I: apply_fun f x ∈ I3 ∩ I.
L174554
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun f x0 ∈ I3 ∩ I) x HxC).
L174554
We prove the intermediate claim HfxI3: apply_fun f x ∈ I3.
L174556
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun f x) HfxI3I).
L174556
We prove the intermediate claim H13R: one_third ∈ R.
L174558
An exact proof term for the current goal is one_third_in_R.
L174558
We prove the intermediate claim H23R: two_thirds ∈ R.
L174560
An exact proof term for the current goal is two_thirds_in_R.
L174560
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L174562
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L174562
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L174564
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L174564
We prove the intermediate claim HfxR: apply_fun f x ∈ R.
L174566
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun f x) HfxI3).
L174566
We prove the intermediate claim Hfx_bounds: Rle one_third (apply_fun f x) ∧ Rle (apply_fun f x) 1.
L174568
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun f x) H13R real_1 HfxI3).
L174568
We prove the intermediate claim H13lefx: Rle one_third (apply_fun f x).
L174570
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun f x)) (Rle (apply_fun f x) 1) Hfx_bounds).
L174570
We prove the intermediate claim Hfxle1: Rle (apply_fun f x) 1.
L174572
An exact proof term for the current goal is (andER (Rle one_third (apply_fun f x)) (Rle (apply_fun f x) 1) Hfx_bounds).
L174572
We prove the intermediate claim Hf1R: add_SNo (apply_fun f x) (minus_SNo one_third) ∈ R.
L174574
An exact proof term for the current goal is (real_add_SNo (apply_fun f x) HfxR (minus_SNo one_third) Hm13R).
L174574
We prove the intermediate claim H0le_f1_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun f x) (minus_SNo one_third)).
L174577
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun f x) (minus_SNo one_third) H13R HfxR Hm13R H13lefx).
L174577
We prove the intermediate claim H13S: SNo one_third.
L174579
An exact proof term for the current goal is (real_SNo one_third H13R).
L174579
We prove the intermediate claim H0le_f1: Rle 0 (add_SNo (apply_fun f x) (minus_SNo one_third)).
L174581
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L174581
An exact proof term for the current goal is H0le_f1_tmp.
L174582
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L174584
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L174584
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun f x) (minus_SNo one_third)).
L174586
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun f x) (minus_SNo one_third)) Hm23le0 H0le_f1).
L174586
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun f x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L174589
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun f x) 1 (minus_SNo one_third) HfxR real_1 Hm13R Hfxle1).
L174589
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun f x) (minus_SNo one_third)) two_thirds.
L174591
rewrite the current goal using two_thirds_eq_1_minus_one_third (from left to right) at position 1.
L174591
An exact proof term for the current goal is Hup_tmp.
L174592
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun f x) (minus_SNo one_third)) Hm23R H23R Hf1R Hlow Hup).
L174596
Assume HxnotC: ¬ (x ∈ C).
L174596
We prove the intermediate claim HnotI1: ¬ (apply_fun f x ∈ I1).
L174598
Assume HfxI1: apply_fun f x ∈ I1.
L174598
We prove the intermediate claim HfxI1I: apply_fun f x ∈ I1 ∩ I.
L174600
An exact proof term for the current goal is (binintersectI I1 I (apply_fun f x) HfxI1 HfxI).
L174600
We prove the intermediate claim HxB': x ∈ B.
L174602
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun f x0 ∈ I1 ∩ I) x HxA HfxI1I).
L174602
Apply FalseE to the current goal.
L174603
An exact proof term for the current goal is (HxnotB HxB').
L174604
We prove the intermediate claim HnotI3: ¬ (apply_fun f x ∈ I3).
L174606
Assume HfxI3: apply_fun f x ∈ I3.
L174606
We prove the intermediate claim HfxI3I: apply_fun f x ∈ I3 ∩ I.
L174608
An exact proof term for the current goal is (binintersectI I3 I (apply_fun f x) HfxI3 HfxI).
L174608
We prove the intermediate claim HxC': x ∈ C.
L174610
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun f x0 ∈ I3 ∩ I) x HxA HfxI3I).
L174610
Apply FalseE to the current goal.
L174611
An exact proof term for the current goal is (HxnotC HxC').
L174612
We prove the intermediate claim HfxR: apply_fun f x ∈ R.
L174614
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun f x) HfxI).
L174614
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L174616
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L174616
We prove the intermediate claim H13R: one_third ∈ R.
L174618
An exact proof term for the current goal is one_third_in_R.
L174618
We prove the intermediate claim Hm13R: (minus_SNo one_third) ∈ R.
L174620
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L174620
We prove the intermediate claim Hfx_bounds: Rle (minus_SNo 1) (apply_fun f x) ∧ Rle (apply_fun f x) 1.
L174622
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun f x) Hm1R real_1 HfxI).
L174622
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun f x) (minus_SNo 1)).
L174624
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun f x) (andEL (Rle (minus_SNo 1) (apply_fun f x)) (Rle (apply_fun f x) 1) Hfx_bounds)).
L174625
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun f x)).
L174627
An exact proof term for the current goal is (RleE_nlt (apply_fun f x) 1 (andER (Rle (minus_SNo 1) (apply_fun f x)) (Rle (apply_fun f x) 1) Hfx_bounds)).
L174628
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun f x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun f x).
L174630
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun f x) Hm1R Hm13R HfxR HnotI1).
L174631
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun f x).
L174633
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun f x))) to the current goal.
L174634
Assume Hbad: Rlt (apply_fun f x) (minus_SNo 1).
L174634
Apply FalseE to the current goal.
L174635
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L174637
Assume Hok: Rlt (minus_SNo one_third) (apply_fun f x).
L174637
An exact proof term for the current goal is Hok.
L174638
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun f x) (minus_SNo one_third)).
L174640
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun f x) Hm13lt_fx).
L174640
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun f x) one_third ∨ Rlt 1 (apply_fun f x).
L174642
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun f x) H13R real_1 HfxR HnotI3).
L174643
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun f x) one_third.
L174645
Apply (HnotI3_cases (Rlt (apply_fun f x) one_third)) to the current goal.
L174646
Assume Hok: Rlt (apply_fun f x) one_third.
L174646
An exact proof term for the current goal is Hok.
L174648
Assume Hbad: Rlt 1 (apply_fun f x).
L174648
Apply FalseE to the current goal.
L174649
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L174650
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun f x)).
L174652
An exact proof term for the current goal is (not_Rlt_sym (apply_fun f x) one_third Hfx_lt_13).
L174652
We prove the intermediate claim HfxI0: apply_fun f x ∈ I0.
L174654
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L174656
We prove the intermediate claim HxSep: apply_fun f x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L174658
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun f x) HfxR (andI (¬ (Rlt (apply_fun f x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun f x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L174662
rewrite the current goal using HI0_def (from left to right).
L174663
An exact proof term for the current goal is HxSep.
L174664
rewrite the current goal using (Hf1_apply x HxA) (from left to right).
L174665
We prove the intermediate claim Hg0xI0: apply_fun g0 x ∈ I0.
L174667
An exact proof term for the current goal is (Hfung0 x HxX).
L174667
We prove the intermediate claim Hg0xR: apply_fun g0 x ∈ R.
L174669
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun g0 x) Hg0xI0).
L174669
We prove the intermediate claim Hm_g0x_R: minus_SNo (apply_fun g0 x) ∈ R.
L174671
An exact proof term for the current goal is (real_minus_SNo (apply_fun g0 x) Hg0xR).
L174671
We prove the intermediate claim Hf1xR: add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x)) ∈ R.
L174673
An exact proof term for the current goal is (real_add_SNo (apply_fun f x) HfxR (minus_SNo (apply_fun g0 x)) Hm_g0x_R).
L174673
We prove the intermediate claim Hm13R: (minus_SNo one_third) ∈ R.
L174675
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L174675
We prove the intermediate claim H23R: two_thirds ∈ R.
L174677
An exact proof term for the current goal is two_thirds_in_R.
L174677
We prove the intermediate claim Hm23R: (minus_SNo two_thirds) ∈ R.
L174679
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L174679
We prove the intermediate claim Hfx0_bounds: Rle (minus_SNo one_third) (apply_fun f x) ∧ Rle (apply_fun f x) one_third.
L174681
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun f x) Hm13R H13R HfxI0).
L174681
We prove the intermediate claim Hg0x_bounds: Rle (minus_SNo one_third) (apply_fun g0 x) ∧ Rle (apply_fun g0 x) one_third.
L174683
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun g0 x) Hm13R H13R Hg0xI0).
L174683
We prove the intermediate claim Hm13_le_fx: Rle (minus_SNo one_third) (apply_fun f x).
L174685
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun f x)) (Rle (apply_fun f x) one_third) Hfx0_bounds).
L174685
We prove the intermediate claim Hfx_le_13: Rle (apply_fun f x) one_third.
L174687
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun f x)) (Rle (apply_fun f x) one_third) Hfx0_bounds).
L174687
We prove the intermediate claim Hm13_le_g0x: Rle (minus_SNo one_third) (apply_fun g0 x).
L174689
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun g0 x)) (Rle (apply_fun g0 x) one_third) Hg0x_bounds).
L174689
We prove the intermediate claim Hg0x_le_13: Rle (apply_fun g0 x) one_third.
L174691
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun g0 x)) (Rle (apply_fun g0 x) one_third) Hg0x_bounds).
L174691
We prove the intermediate claim Hm13_le_mg0x: Rle (minus_SNo one_third) (minus_SNo (apply_fun g0 x)).
L174693
An exact proof term for the current goal is (Rle_minus_contra (apply_fun g0 x) one_third Hg0x_le_13).
L174693
We prove the intermediate claim Hmg0x_le_13: Rle (minus_SNo (apply_fun g0 x)) one_third.
L174695
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun g0 x)) (minus_SNo (minus_SNo one_third)).
L174696
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun g0 x) Hm13_le_g0x).
L174696
We prove the intermediate claim H13R: one_third ∈ R.
L174698
An exact proof term for the current goal is one_third_in_R.
L174698
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L174699
An exact proof term for the current goal is Htmp.
L174700
We prove the intermediate claim Hlo1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun f x) (minus_SNo one_third)).
L174703
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun f x) (minus_SNo one_third) Hm13R HfxR Hm13R Hm13_le_fx).
L174704
We prove the intermediate claim Hlo2: Rle (add_SNo (apply_fun f x) (minus_SNo one_third)) (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))).
L174707
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun f x) (minus_SNo one_third) (minus_SNo (apply_fun g0 x)) HfxR Hm13R Hm_g0x_R Hm13_le_mg0x).
L174708
We prove the intermediate claim Hlo': Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))).
L174711
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun f x) (minus_SNo one_third)) (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))) Hlo1 Hlo2).
L174714
We prove the intermediate claim Hlo: Rle (minus_SNo two_thirds) (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))).
L174716
rewrite the current goal using (minus_two_thirds_eq) (from left to right) at position 1.
L174716
An exact proof term for the current goal is Hlo'.
L174717
We prove the intermediate claim Hhi1: Rle (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))) (add_SNo one_third (minus_SNo (apply_fun g0 x))).
L174720
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun f x) one_third (minus_SNo (apply_fun g0 x)) HfxR H13R Hm_g0x_R Hfx_le_13).
L174721
We prove the intermediate claim Hhi2: Rle (add_SNo one_third (minus_SNo (apply_fun g0 x))) (add_SNo one_third one_third).
L174724
An exact proof term for the current goal is (Rle_add_SNo_2 one_third (minus_SNo (apply_fun g0 x)) one_third H13R Hm_g0x_R H13R Hmg0x_le_13).
L174725
We prove the intermediate claim Hhi': Rle (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))) (add_SNo one_third one_third).
L174728
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))) (add_SNo one_third (minus_SNo (apply_fun g0 x))) (add_SNo one_third one_third) Hhi1 Hhi2).
L174731
We prove the intermediate claim Hhi: Rle (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))) two_thirds.
L174733
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L174734
rewrite the current goal using Hdef23 (from left to right).
L174735
An exact proof term for the current goal is Hhi'.
L174736
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))) Hm23R H23R Hf1xR Hlo Hhi).
L174739
We prove the intermediate claim Hseries: ∃gR : set, continuous_map X Tx R R_standard_topology gR ∧ (∀x : set, x ∈ A → apply_fun gR x = apply_fun f x) ∧ (∀x : set, x ∈ X → apply_fun gR x ∈ closed_interval (minus_SNo 1) 1).
(*** Further iteration and summation still pending. ***)
L174746
Set I to be the term closed_interval (minus_SNo 1) 1.
L174747
Set Ti to be the term closed_interval_topology (minus_SNo 1) 1.
(*** TeX Step II: geometric series of successive Step I corrections. ***)
L174748
We prove the intermediate claim H23R: two_thirds ∈ R.
L174750
An exact proof term for the current goal is two_thirds_in_R.
L174750
We prove the intermediate claim H23S: SNo two_thirds.
L174752
An exact proof term for the current goal is (real_SNo two_thirds H23R).
L174752
We prove the intermediate claim H23pos: 0 < two_thirds.
L174754
An exact proof term for the current goal is two_thirds_pos.
L174754
We prove the intermediate claim H23ne0: two_thirds ≠ 0.
L174756
An exact proof term for the current goal is two_thirds_ne0.
L174756
Set den to be the term two_thirds.
L174759
Set f1s to be the term compose_fun A f1 (div_const_fun den).
(*** define scaled residual on A: f1s = f1 / (2/3), so f1s maps into [-1,1] ***)
L174760
We prove the intermediate claim Hdivfun: function_on (div_const_fun den) R R.
L174762
Let t be given.
L174762
Assume HtR: t ∈ R.
L174762
An exact proof term for the current goal is (div_const_fun_value_in_R den t H23R HtR).
L174763
We prove the intermediate claim Hf1fun0: function_on f1 A R.
L174765
An exact proof term for the current goal is (andEL (function_on f1 A R) (∀a : set, a ∈ A → ∃y : set, y ∈ R ∧ (a,y) ∈ f1) Hf1_total).
L174767
We prove the intermediate claim Hf1s_total: total_function_on f1s A R.
L174769
An exact proof term for the current goal is (total_function_on_compose_fun A R R f1 (div_const_fun den) Hf1fun0 Hdivfun).
L174769
We prove the intermediate claim Hf1s_fun: function_on f1s A R.
L174771
An exact proof term for the current goal is (andEL (function_on f1s A R) (∀x : set, x ∈ A → ∃y : set, y ∈ R ∧ (x,y) ∈ f1s) Hf1s_total).
L174773
We prove the intermediate claim Hf1s_apply: ∀x : set, x ∈ A → apply_fun f1s x = div_SNo (apply_fun f1 x) den.
L174776
Let x be given.
L174776
Assume HxA: x ∈ A.
L174776
We prove the intermediate claim HxR: apply_fun f1 x ∈ R.
L174778
An exact proof term for the current goal is (Hf1fun0 x HxA).
L174778
rewrite the current goal using (compose_fun_apply A f1 (div_const_fun den) x HxA) (from left to right).
L174779
rewrite the current goal using (div_const_fun_apply den (apply_fun f1 x) H23R HxR) (from left to right).
Use reflexivity.
L174781
We prove the intermediate claim Hf1s_I: ∀x : set, x ∈ A → apply_fun f1s x ∈ I.
(*** pending: prove f1s is continuous as a map A -> I with the interval topology ***)
L174785
Let x be given.
L174785
Assume HxA: x ∈ A.
L174785
We will prove apply_fun f1s x ∈ I.
L174786
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L174788
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L174788
We prove the intermediate claim Hf1xI2: apply_fun f1 x ∈ I2.
L174791
An exact proof term for the current goal is (Hf1_range x HxA).
L174791
We prove the intermediate claim Hm23R: (minus_SNo den) ∈ R.
L174793
An exact proof term for the current goal is (real_minus_SNo den H23R).
L174793
We prove the intermediate claim Hf1xR: apply_fun f1 x ∈ R.
L174795
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun f1 x) Hf1xI2).
L174795
We prove the intermediate claim Hf1xS: SNo (apply_fun f1 x).
L174797
An exact proof term for the current goal is (real_SNo (apply_fun f1 x) Hf1xR).
L174797
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun f1 x) ∧ Rle (apply_fun f1 x) den.
L174800
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun f1 x) Hm23R H23R Hf1xI2).
L174800
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun f1 x).
L174802
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun f1 x)) (Rle (apply_fun f1 x) den) Hbounds).
L174804
We prove the intermediate claim Hhi: Rle (apply_fun f1 x) den.
L174806
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun f1 x)) (Rle (apply_fun f1 x) den) Hbounds).
L174808
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun f1 x)).
L174810
An exact proof term for the current goal is (RleE_nlt (apply_fun f1 x) den Hhi).
L174810
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun f1 x) (minus_SNo den)).
L174812
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun f1 x) Hlo).
L174812
We prove the intermediate claim HyEq: apply_fun f1s x = div_SNo (apply_fun f1 x) den.
L174815
An exact proof term for the current goal is (Hf1s_apply x HxA).
L174815
We prove the intermediate claim HyR: apply_fun f1s x ∈ R.
L174817
rewrite the current goal using HyEq (from left to right).
L174817
An exact proof term for the current goal is (real_div_SNo (apply_fun f1 x) Hf1xR den H23R).
L174818
We prove the intermediate claim HyS: SNo (apply_fun f1s x).
L174820
An exact proof term for the current goal is (real_SNo (apply_fun f1s x) HyR).
L174820
We prove the intermediate claim Hy_le_1: Rle (apply_fun f1s x) 1.
L174823
Apply (RleI (apply_fun f1s x) 1 HyR real_1) to the current goal.
L174823
We will prove ¬ (Rlt 1 (apply_fun f1s x)).
L174824
Assume H1lt: Rlt 1 (apply_fun f1s x).
L174825
We prove the intermediate claim H1lty: 1 < apply_fun f1s x.
L174827
An exact proof term for the current goal is (RltE_lt 1 (apply_fun f1s x) H1lt).
L174827
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun f1s x) den.
L174829
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun f1s x) den SNo_1 HyS H23S H23pos H1lty).
L174829
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun f1s x) den.
L174831
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L174831
An exact proof term for the current goal is HmulLt.
L174832
We prove the intermediate claim HmulEq: mul_SNo (apply_fun f1s x) den = apply_fun f1 x.
L174834
rewrite the current goal using HyEq (from left to right) at position 1.
L174834
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun f1 x) den Hf1xS H23S H23ne0).
L174835
We prove the intermediate claim Hden_lt_f1x: den < apply_fun f1 x.
L174837
rewrite the current goal using HmulEq (from right to left).
L174837
An exact proof term for the current goal is HmulLt'.
L174838
We prove the intermediate claim Hbad: Rlt den (apply_fun f1 x).
L174840
An exact proof term for the current goal is (RltI den (apply_fun f1 x) H23R Hf1xR Hden_lt_f1x).
L174840
An exact proof term for the current goal is (Hnlt_hi Hbad).
L174841
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun f1s x).
L174844
Apply (RleI (minus_SNo 1) (apply_fun f1s x) Hm1R HyR) to the current goal.
L174844
We will prove ¬ (Rlt (apply_fun f1s x) (minus_SNo 1)).
L174845
Assume Hylt: Rlt (apply_fun f1s x) (minus_SNo 1).
L174846
We prove the intermediate claim Hylts: apply_fun f1s x < minus_SNo 1.
L174848
An exact proof term for the current goal is (RltE_lt (apply_fun f1s x) (minus_SNo 1) Hylt).
L174848
We prove the intermediate claim HmulLt: mul_SNo (apply_fun f1s x) den < mul_SNo (minus_SNo 1) den.
L174850
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun f1s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L174851
We prove the intermediate claim HmulEq: mul_SNo (apply_fun f1s x) den = apply_fun f1 x.
L174853
rewrite the current goal using HyEq (from left to right) at position 1.
L174853
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun f1 x) den Hf1xS H23S H23ne0).
L174854
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L174856
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L174856
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L174858
We prove the intermediate claim Hf1x_lt_mden: apply_fun f1 x < minus_SNo den.
L174860
rewrite the current goal using HmulEq (from right to left).
L174860
rewrite the current goal using HrhsEq (from right to left).
L174861
An exact proof term for the current goal is HmulLt.
L174862
We prove the intermediate claim Hbad: Rlt (apply_fun f1 x) (minus_SNo den).
L174864
An exact proof term for the current goal is (RltI (apply_fun f1 x) (minus_SNo den) Hf1xR Hm23R Hf1x_lt_mden).
L174864
An exact proof term for the current goal is (Hnlt_lo Hbad).
L174865
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun f1s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L174867
We prove the intermediate claim Hf1s_cont: continuous_map A (subspace_topology X Tx A) I Ti f1s.
L174869
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
(*** reduce to continuity into R and range restriction to I ***)
L174871
An exact proof term for the current goal is R_standard_topology_is_topology.
L174871
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L174873
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L174873
We prove the intermediate claim Hf1cont: continuous_map A (subspace_topology X Tx A) R R_standard_topology f1.
L174875
Set Ta to be the term subspace_topology X Tx A.
L174876
We prove the intermediate claim HTa: topology_on A Ta.
(*** f1 is composition of pair_map with add_fun_R ***)
L174878
An exact proof term for the current goal is (subspace_topology_is_topology X Tx A HTx HAsubX).
L174878
We prove the intermediate claim HIcR: I ⊆ R.
L174880
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1).
L174880
We prove the intermediate claim HTiEq: (closed_interval_topology (minus_SNo 1) 1) = subspace_topology R R_standard_topology I.
Use reflexivity.
L174883
We prove the intermediate claim HfcontR: continuous_map A Ta R R_standard_topology f.
L174885
An exact proof term for the current goal is (continuous_map_range_expand A Ta I (closed_interval_topology (minus_SNo 1) 1) R R_standard_topology f Hf HIcR R_standard_topology_is_topology_local HTiEq).
L174890
We prove the intermediate claim Hg0contA: continuous_map A Ta R R_standard_topology g0.
L174892
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology g0 A HTx HAsubX Hg0contR).
L174892
We prove the intermediate claim Hnegcont: continuous_map R R_standard_topology R R_standard_topology neg_fun.
L174894
An exact proof term for the current goal is neg_fun_continuous.
L174894
Set g0neg to be the term compose_fun A g0 neg_fun.
L174895
We prove the intermediate claim Hg0negcont: continuous_map A Ta R R_standard_topology g0neg.
L174897
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology g0 neg_fun Hg0contA Hnegcont).
L174898
Set h to be the term pair_map A f g0neg.
L174899
We prove the intermediate claim Hhcont: continuous_map A Ta (setprod R R) (product_topology R R_standard_topology R R_standard_topology) h.
L174903
An exact proof term for the current goal is (maps_into_products_axiom A Ta R R_standard_topology R R_standard_topology f g0neg HfcontR Hg0negcont).
L174904
We prove the intermediate claim Haddcont: continuous_map (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology add_fun_R.
L174909
An exact proof term for the current goal is add_fun_R_continuous.
L174909
Set f1c to be the term compose_fun A h add_fun_R.
L174910
We prove the intermediate claim Hf1c_cont: continuous_map A Ta R R_standard_topology f1c.
L174912
An exact proof term for the current goal is (composition_continuous A Ta (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology h add_fun_R Hhcont Haddcont).
L174915
We prove the intermediate claim Heq: f1 = f1c.
L174917
Apply set_ext to the current goal.
L174918
Let p be given.
L174918
Assume Hp: p ∈ f1.
L174918
We will prove p ∈ f1c.
L174919
Apply (ReplE_impred A (λx0 : set ⇒ (x0,add_SNo (apply_fun f x0) (minus_SNo (apply_fun g0 x0)))) p Hp (p ∈ f1c)) to the current goal.
L174924
Let x be given.
L174925
Assume HxA: x ∈ A.
L174925
Assume Hpeq: p = (x,add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))).
L174926
rewrite the current goal using Hpeq (from left to right).
L174927
We prove the intermediate claim Hhx: apply_fun h x = (apply_fun f x,apply_fun g0neg x).
L174929
An exact proof term for the current goal is (pair_map_apply A R R f g0neg x HxA).
L174929
We prove the intermediate claim Hg0negx: apply_fun g0neg x = minus_SNo (apply_fun g0 x).
L174931
We prove the intermediate claim HxTa: x ∈ A.
L174932
An exact proof term for the current goal is HxA.
L174932
We prove the intermediate claim HxX: x ∈ X.
L174934
An exact proof term for the current goal is (HAsubX x HxA).
L174934
We prove the intermediate claim Hg0xR: apply_fun g0 x ∈ R.
L174936
An exact proof term for the current goal is (continuous_map_function_on X Tx R R_standard_topology g0 Hg0contR x HxX).
L174936
rewrite the current goal using (compose_fun_apply A g0 neg_fun x HxA) (from left to right) at position 1.
L174937
rewrite the current goal using (neg_fun_apply (apply_fun g0 x) Hg0xR) (from left to right) at position 1.
Use reflexivity.
L174939
We prove the intermediate claim Hhimg: apply_fun h x ∈ setprod R R.
L174941
An exact proof term for the current goal is (continuous_map_function_on A Ta (setprod R R) (product_topology R R_standard_topology R R_standard_topology) h Hhcont x HxA).
L174942
We prove the intermediate claim Happ: apply_fun f1c x = apply_fun add_fun_R (apply_fun h x).
L174944
An exact proof term for the current goal is (compose_fun_apply A h add_fun_R x HxA).
L174944
We prove the intermediate claim Hadd: apply_fun add_fun_R (apply_fun h x) = add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x)).
L174947
rewrite the current goal using (add_fun_R_apply (apply_fun h x) Hhimg) (from left to right) at position 1.
L174947
rewrite the current goal using Hhx (from left to right).
L174948
rewrite the current goal using (tuple_2_0_eq (apply_fun f x) (apply_fun g0neg x)) (from left to right).
L174949
rewrite the current goal using (tuple_2_1_eq (apply_fun f x) (apply_fun g0neg x)) (from left to right).
L174950
rewrite the current goal using Hg0negx (from left to right) at position 1.
Use reflexivity.
L174952
rewrite the current goal using Hadd (from right to left).
L174953
An exact proof term for the current goal is (ReplI A (λx0 : set ⇒ (x0,apply_fun add_fun_R (apply_fun h x0))) x HxA).
L174955
Let p be given.
L174955
Assume Hp: p ∈ f1c.
L174955
We will prove p ∈ f1.
L174956
Apply (ReplE_impred A (λx0 : set ⇒ (x0,apply_fun add_fun_R (apply_fun h x0))) p Hp (p ∈ f1)) to the current goal.
L174957
Let x be given.
L174958
Assume HxA: x ∈ A.
L174958
Assume Hpeq: p = (x,apply_fun add_fun_R (apply_fun h x)).
L174959
rewrite the current goal using Hpeq (from left to right).
L174960
We prove the intermediate claim Hhx: apply_fun h x = (apply_fun f x,apply_fun g0neg x).
L174962
An exact proof term for the current goal is (pair_map_apply A R R f g0neg x HxA).
L174962
We prove the intermediate claim Hg0xR: apply_fun g0 x ∈ R.
L174964
We prove the intermediate claim HxX: x ∈ X.
L174965
An exact proof term for the current goal is (HAsubX x HxA).
L174965
An exact proof term for the current goal is (continuous_map_function_on X Tx R R_standard_topology g0 Hg0contR x HxX).
L174966
We prove the intermediate claim Hg0negx: apply_fun g0neg x = minus_SNo (apply_fun g0 x).
L174968
rewrite the current goal using (compose_fun_apply A g0 neg_fun x HxA) (from left to right) at position 1.
L174968
rewrite the current goal using (neg_fun_apply (apply_fun g0 x) Hg0xR) (from left to right) at position 1.
Use reflexivity.
L174970
We prove the intermediate claim Hadd: apply_fun add_fun_R (apply_fun h x) = add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x)).
L174973
We prove the intermediate claim Hhimg: apply_fun h x ∈ setprod R R.
L174974
An exact proof term for the current goal is (continuous_map_function_on A Ta (setprod R R) (product_topology R R_standard_topology R R_standard_topology) h Hhcont x HxA).
L174975
rewrite the current goal using (add_fun_R_apply (apply_fun h x) Hhimg) (from left to right) at position 1.
L174976
rewrite the current goal using Hhx (from left to right).
L174977
rewrite the current goal using (tuple_2_0_eq (apply_fun f x) (apply_fun g0neg x)) (from left to right).
L174978
rewrite the current goal using (tuple_2_1_eq (apply_fun f x) (apply_fun g0neg x)) (from left to right).
L174979
rewrite the current goal using Hg0negx (from left to right) at position 1.
Use reflexivity.
L174981
rewrite the current goal using Hadd (from left to right).
L174982
An exact proof term for the current goal is (ReplI A (λx0 : set ⇒ (x0,add_SNo (apply_fun f x0) (minus_SNo (apply_fun g0 x0)))) x HxA).
L174986
rewrite the current goal using Heq (from left to right).
L174987
An exact proof term for the current goal is Hf1c_cont.
L174988
We prove the intermediate claim Hf1s_cont_R: continuous_map A (subspace_topology X Tx A) R R_standard_topology f1s.
L174990
An exact proof term for the current goal is (composition_continuous A (subspace_topology X Tx A) R R_standard_topology R R_standard_topology f1 (div_const_fun den) Hf1cont Hdivcont).
L174991
We prove the intermediate claim HISubR: I ⊆ R.
L174993
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1).
L174993
We prove the intermediate claim HTiEq: Ti = subspace_topology R R_standard_topology I.
Use reflexivity.
L174995
rewrite the current goal using HTiEq (from left to right).
L174996
An exact proof term for the current goal is (continuous_map_range_restrict A (subspace_topology X Tx A) R R_standard_topology f1s I Hf1s_cont_R HISubR Hf1s_I).
L174998
We prove the intermediate claim Hex_u1: ∃u1 : set, continuous_map X Tx (closed_interval (minus_SNo one_third) one_third) (closed_interval_topology (minus_SNo one_third) one_third) u1 ∧ (∀x : set, x ∈ preimage_of A f1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u1 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A f1s ((closed_interval one_third 1) ∩ I) → apply_fun u1 x = one_third).
(*** apply Step I to f1s to obtain the next correction u1:X -> [-1/3,1/3] ***)
L175008
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A f1s Hnorm HA Hf1s_cont).
L175008
Apply Hex_u1 to the current goal.
L175009
Let u1 be given.
L175010
Assume Hu1.
L175010
Set I0 to be the term closed_interval (minus_SNo one_third) one_third.
L175011
Set T0 to be the term closed_interval_topology (minus_SNo one_third) one_third.
L175012
We prove the intermediate claim Hu1AB: continuous_map X Tx I0 T0 u1 ∧ (∀x : set, x ∈ preimage_of A f1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u1 x = minus_SNo one_third).
L175017
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u1 ∧ (∀x : set, x ∈ preimage_of A f1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u1 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A f1s ((closed_interval one_third 1) ∩ I) → apply_fun u1 x = one_third) Hu1).
L175023
We prove the intermediate claim Hu1contI0: continuous_map X Tx I0 T0 u1.
L175025
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u1) (∀x : set, x ∈ preimage_of A f1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u1 x = minus_SNo one_third) Hu1AB).
L175029
We prove the intermediate claim Hu1contR: continuous_map X Tx R R_standard_topology u1.
L175031
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L175032
We prove the intermediate claim HI0subR: I0 ⊆ R.
L175034
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L175034
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u1 Hu1contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L175042
Set den to be the term two_thirds.
L175044
Set u1s to be the term compose_fun X u1 (mul_const_fun den).
(*** a first partial sum g1 = g0 + (2/3) u1 is continuous ***)
L175045
We prove the intermediate claim HdenR: den ∈ R.
L175047
An exact proof term for the current goal is two_thirds_in_R.
L175047
We prove the intermediate claim HdenPos: 0 < den.
L175049
An exact proof term for the current goal is two_thirds_pos.
L175049
We prove the intermediate claim HmulCont: continuous_map R R_standard_topology R R_standard_topology (mul_const_fun den).
L175051
An exact proof term for the current goal is (mul_const_fun_continuous_pos den HdenR HdenPos).
L175051
We prove the intermediate claim Hu1s_cont: continuous_map X Tx R R_standard_topology u1s.
L175053
An exact proof term for the current goal is (composition_continuous X Tx R R_standard_topology R R_standard_topology u1 (mul_const_fun den) Hu1contR HmulCont).
L175054
Set h1 to be the term pair_map X g0 u1s.
L175055
We prove the intermediate claim Hh1cont: continuous_map X Tx (setprod R R) (product_topology R R_standard_topology R R_standard_topology) h1.
L175059
An exact proof term for the current goal is (maps_into_products_axiom X Tx R R_standard_topology R R_standard_topology g0 u1s Hg0contR Hu1s_cont).
L175060
We prove the intermediate claim Haddcont: continuous_map (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology add_fun_R.
L175065
An exact proof term for the current goal is add_fun_R_continuous.
L175065
Set g1 to be the term compose_fun X h1 add_fun_R.
L175066
We prove the intermediate claim Hg1cont: continuous_map X Tx R R_standard_topology g1.
L175068
An exact proof term for the current goal is (composition_continuous X Tx (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology h1 add_fun_R Hh1cont Haddcont).
L175071
Set Ta to be the term subspace_topology X Tx A.
L175074
We prove the intermediate claim HTa: topology_on A Ta.
(*** second correction step: build the next residual on A and a second partial sum ***)
L175076
An exact proof term for the current goal is (subspace_topology_is_topology X Tx A HTx HAsubX).
L175076
We prove the intermediate claim HIcR: I ⊆ R.
L175078
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1).
L175078
We prove the intermediate claim HTiEq: Ti = subspace_topology R R_standard_topology I.
Use reflexivity.
L175080
We prove the intermediate claim Hf1s_contR: continuous_map A Ta R R_standard_topology f1s.
L175082
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology f1s Hf1s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L175083
We prove the intermediate claim Hu1contA: continuous_map A Ta R R_standard_topology u1.
L175085
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u1 A HTx HAsubX Hu1contR).
L175085
We prove the intermediate claim Hnegcont: continuous_map R R_standard_topology R R_standard_topology neg_fun.
L175087
An exact proof term for the current goal is neg_fun_continuous.
L175087
Set u1neg to be the term compose_fun A u1 neg_fun.
L175088
We prove the intermediate claim Hu1neg_cont: continuous_map A Ta R R_standard_topology u1neg.
L175090
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u1 neg_fun Hu1contA Hnegcont).
L175091
Set r1 to be the term compose_fun A (pair_map A f1s u1neg) add_fun_R.
L175092
We prove the intermediate claim Hr1_cont: continuous_map A Ta R R_standard_topology r1.
L175094
An exact proof term for the current goal is (add_two_continuous_R A Ta f1s u1neg HTa Hf1s_contR Hu1neg_cont).
L175094
We prove the intermediate claim Hr1_apply: ∀x : set, x ∈ A → apply_fun r1 x = add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x)).
L175097
Let x be given.
L175097
Assume HxA: x ∈ A.
L175097
We prove the intermediate claim Hpimg: apply_fun (pair_map A f1s u1neg) x ∈ setprod R R.
L175099
rewrite the current goal using (pair_map_apply A R R f1s u1neg x HxA) (from left to right).
L175099
We prove the intermediate claim Hf1sRx: apply_fun f1s x ∈ R.
L175101
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology f1s Hf1s_contR x HxA).
L175101
We prove the intermediate claim Hu1negRx: apply_fun u1neg x ∈ R.
L175103
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u1neg Hu1neg_cont x HxA).
L175103
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun f1s x) (apply_fun u1neg x) Hf1sRx Hu1negRx).
L175104
rewrite the current goal using (compose_fun_apply A (pair_map A f1s u1neg) add_fun_R x HxA) (from left to right).
L175105
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A f1s u1neg) x) Hpimg) (from left to right) at position 1.
L175106
rewrite the current goal using (pair_map_apply A R R f1s u1neg x HxA) (from left to right).
L175107
rewrite the current goal using (tuple_2_0_eq (apply_fun f1s x) (apply_fun u1neg x)) (from left to right).
L175108
rewrite the current goal using (tuple_2_1_eq (apply_fun f1s x) (apply_fun u1neg x)) (from left to right).
L175109
rewrite the current goal using (compose_fun_apply A u1 neg_fun x HxA) (from left to right) at position 1.
L175110
We prove the intermediate claim Hu1Rx: apply_fun u1 x ∈ R.
L175112
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u1 Hu1contA x HxA).
L175112
rewrite the current goal using (neg_fun_apply (apply_fun u1 x) Hu1Rx) (from left to right) at position 1.
Use reflexivity.
L175114
We prove the intermediate claim Hu1_on_B1: ∀x : set, x ∈ preimage_of A f1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u1 x = minus_SNo one_third.
(*** show the next residual stays within [-2/3,2/3] on A ***)
L175120
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u1) (∀x : set, x ∈ preimage_of A f1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u1 x = minus_SNo one_third) Hu1AB).
L175124
We prove the intermediate claim Hu1_on_C1: ∀x : set, x ∈ preimage_of A f1s ((closed_interval one_third 1) ∩ I) → apply_fun u1 x = one_third.
L175128
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u1 ∧ (∀x : set, x ∈ preimage_of A f1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u1 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A f1s ((closed_interval one_third 1) ∩ I) → apply_fun u1 x = one_third) Hu1).
L175134
We prove the intermediate claim Hr1_range: ∀x : set, x ∈ A → apply_fun r1 x ∈ I2.
L175136
Let x be given.
L175136
Assume HxA: x ∈ A.
L175136
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L175137
Set I3 to be the term closed_interval one_third 1.
L175138
Set B1 to be the term preimage_of A f1s (I1 ∩ I).
L175139
Set C1 to be the term preimage_of A f1s (I3 ∩ I).
L175140
We prove the intermediate claim Hf1sIx: apply_fun f1s x ∈ I.
L175142
An exact proof term for the current goal is (Hf1s_I x HxA).
L175142
We prove the intermediate claim HB1_cases: x ∈ B1 ∨ ¬ (x ∈ B1).
L175144
An exact proof term for the current goal is (xm (x ∈ B1)).
L175144
Apply (HB1_cases (apply_fun r1 x ∈ I2)) to the current goal.
L175146
Assume HxB1: x ∈ B1.
L175146
We prove the intermediate claim Hu1eq: apply_fun u1 x = minus_SNo one_third.
L175148
An exact proof term for the current goal is (Hu1_on_B1 x HxB1).
L175148
We prove the intermediate claim Hr1eq: apply_fun r1 x = add_SNo (apply_fun f1s x) one_third.
L175150
rewrite the current goal using (Hr1_apply x HxA) (from left to right).
L175150
rewrite the current goal using Hu1eq (from left to right) at position 1.
L175151
We prove the intermediate claim H13R: one_third ∈ R.
L175153
An exact proof term for the current goal is one_third_in_R.
L175153
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L175155
rewrite the current goal using Hr1eq (from left to right).
L175156
We prove the intermediate claim Hf1sI1I: apply_fun f1s x ∈ I1 ∩ I.
L175158
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun f1s x0 ∈ I1 ∩ I) x HxB1).
L175158
We prove the intermediate claim Hf1sI1: apply_fun f1s x ∈ I1.
L175160
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun f1s x) Hf1sI1I).
L175160
We prove the intermediate claim H13R: one_third ∈ R.
L175162
An exact proof term for the current goal is one_third_in_R.
L175162
We prove the intermediate claim H23R: two_thirds ∈ R.
L175164
An exact proof term for the current goal is two_thirds_in_R.
L175164
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L175166
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L175166
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L175168
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L175168
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L175170
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L175170
We prove the intermediate claim Hf1sRx: apply_fun f1s x ∈ R.
L175172
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun f1s x) Hf1sI1).
L175172
We prove the intermediate claim Hf1s_bounds: Rle (minus_SNo 1) (apply_fun f1s x) ∧ Rle (apply_fun f1s x) (minus_SNo one_third).
L175174
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun f1s x) Hm1R Hm13R Hf1sI1).
L175175
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun f1s x).
L175177
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun f1s x)) (Rle (apply_fun f1s x) (minus_SNo one_third)) Hf1s_bounds).
L175178
We prove the intermediate claim HhiI1: Rle (apply_fun f1s x) (minus_SNo one_third).
L175180
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun f1s x)) (Rle (apply_fun f1s x) (minus_SNo one_third)) Hf1s_bounds).
L175181
We prove the intermediate claim Hr1Rx: add_SNo (apply_fun f1s x) one_third ∈ R.
L175183
An exact proof term for the current goal is (real_add_SNo (apply_fun f1s x) Hf1sRx one_third H13R).
L175183
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun f1s x) one_third).
L175185
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun f1s x) one_third Hm1R Hf1sRx H13R Hm1le).
L175185
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun f1s x) one_third).
L175187
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L175187
An exact proof term for the current goal is Hlow_tmp.
L175188
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun f1s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L175190
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun f1s x) (minus_SNo one_third) one_third Hf1sRx Hm13R H13R HhiI1).
L175190
We prove the intermediate claim H13S: SNo one_third.
L175192
An exact proof term for the current goal is (real_SNo one_third H13R).
L175192
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun f1s x) one_third) 0.
L175194
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L175194
An exact proof term for the current goal is Hup0_tmp.
L175195
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L175197
An exact proof term for the current goal is Rle_0_two_thirds.
L175197
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun f1s x) one_third) two_thirds.
L175199
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun f1s x) one_third) 0 two_thirds Hup0 H0le23).
L175199
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun f1s x) one_third) Hm23R H23R Hr1Rx Hlow Hup).
L175203
Assume HxnotB1: ¬ (x ∈ B1).
L175203
We prove the intermediate claim HC1_cases: x ∈ C1 ∨ ¬ (x ∈ C1).
L175205
An exact proof term for the current goal is (xm (x ∈ C1)).
L175205
Apply (HC1_cases (apply_fun r1 x ∈ I2)) to the current goal.
L175207
Assume HxC1: x ∈ C1.
L175207
We prove the intermediate claim Hu1eq: apply_fun u1 x = one_third.
L175209
An exact proof term for the current goal is (Hu1_on_C1 x HxC1).
L175209
We prove the intermediate claim Hr1eq: apply_fun r1 x = add_SNo (apply_fun f1s x) (minus_SNo one_third).
L175211
rewrite the current goal using (Hr1_apply x HxA) (from left to right).
L175211
rewrite the current goal using Hu1eq (from left to right) at position 1.
Use reflexivity.
L175213
rewrite the current goal using Hr1eq (from left to right).
L175214
We prove the intermediate claim Hf1sI3I: apply_fun f1s x ∈ I3 ∩ I.
L175216
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun f1s x0 ∈ I3 ∩ I) x HxC1).
L175216
We prove the intermediate claim Hf1sI3: apply_fun f1s x ∈ I3.
L175218
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun f1s x) Hf1sI3I).
L175218
We prove the intermediate claim H13R: one_third ∈ R.
L175220
An exact proof term for the current goal is one_third_in_R.
L175220
We prove the intermediate claim H23R: two_thirds ∈ R.
L175222
An exact proof term for the current goal is two_thirds_in_R.
L175222
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L175224
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L175224
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L175226
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L175226
We prove the intermediate claim Hf1sRx: apply_fun f1s x ∈ R.
L175228
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun f1s x) Hf1sI3).
L175228
We prove the intermediate claim Hf1s_bounds: Rle one_third (apply_fun f1s x) ∧ Rle (apply_fun f1s x) 1.
L175230
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun f1s x) H13R real_1 Hf1sI3).
L175230
We prove the intermediate claim HloI3: Rle one_third (apply_fun f1s x).
L175232
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun f1s x)) (Rle (apply_fun f1s x) 1) Hf1s_bounds).
L175232
We prove the intermediate claim HhiI3: Rle (apply_fun f1s x) 1.
L175234
An exact proof term for the current goal is (andER (Rle one_third (apply_fun f1s x)) (Rle (apply_fun f1s x) 1) Hf1s_bounds).
L175234
We prove the intermediate claim Hr1Rx: add_SNo (apply_fun f1s x) (minus_SNo one_third) ∈ R.
L175236
An exact proof term for the current goal is (real_add_SNo (apply_fun f1s x) Hf1sRx (minus_SNo one_third) Hm13R).
L175236
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun f1s x) (minus_SNo one_third)).
L175239
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun f1s x) (minus_SNo one_third) H13R Hf1sRx Hm13R HloI3).
L175240
We prove the intermediate claim H13S: SNo one_third.
L175242
An exact proof term for the current goal is (real_SNo one_third H13R).
L175242
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun f1s x) (minus_SNo one_third)).
L175244
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L175244
An exact proof term for the current goal is H0le_tmp.
L175245
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L175247
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L175247
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun f1s x) (minus_SNo one_third)).
L175249
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun f1s x) (minus_SNo one_third)) Hm23le0 H0le).
L175251
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun f1s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L175254
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun f1s x) 1 (minus_SNo one_third) Hf1sRx real_1 Hm13R HhiI3).
L175255
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L175257
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L175257
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L175258
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun f1s x) (minus_SNo one_third)) two_thirds.
L175260
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun f1s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L175263
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun f1s x) (minus_SNo one_third)) Hm23R H23R Hr1Rx Hlow Hup).
L175267
Assume HxnotC1: ¬ (x ∈ C1).
L175267
We prove the intermediate claim HxX: x ∈ X.
L175269
An exact proof term for the current goal is (HAsubX x HxA).
L175269
We prove the intermediate claim HnotI1: ¬ (apply_fun f1s x ∈ I1).
L175271
Assume Hf1sI1': apply_fun f1s x ∈ I1.
L175271
We prove the intermediate claim Hf1sI1I: apply_fun f1s x ∈ I1 ∩ I.
L175273
An exact proof term for the current goal is (binintersectI I1 I (apply_fun f1s x) Hf1sI1' Hf1sIx).
L175273
We prove the intermediate claim HxB1': x ∈ B1.
L175275
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun f1s x0 ∈ I1 ∩ I) x HxA Hf1sI1I).
L175275
Apply FalseE to the current goal.
L175276
An exact proof term for the current goal is (HxnotB1 HxB1').
L175277
We prove the intermediate claim HnotI3: ¬ (apply_fun f1s x ∈ I3).
L175279
Assume Hf1sI3': apply_fun f1s x ∈ I3.
L175279
We prove the intermediate claim Hf1sI3I: apply_fun f1s x ∈ I3 ∩ I.
L175281
An exact proof term for the current goal is (binintersectI I3 I (apply_fun f1s x) Hf1sI3' Hf1sIx).
L175281
We prove the intermediate claim HxC1': x ∈ C1.
L175283
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun f1s x0 ∈ I3 ∩ I) x HxA Hf1sI3I).
L175283
Apply FalseE to the current goal.
L175284
An exact proof term for the current goal is (HxnotC1 HxC1').
L175285
We prove the intermediate claim Hf1sRx: apply_fun f1s x ∈ R.
L175287
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun f1s x) Hf1sIx).
L175287
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L175289
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L175289
We prove the intermediate claim H13R: one_third ∈ R.
L175291
An exact proof term for the current goal is one_third_in_R.
L175291
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L175293
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L175293
We prove the intermediate claim Hf1s_boundsI: Rle (minus_SNo 1) (apply_fun f1s x) ∧ Rle (apply_fun f1s x) 1.
L175295
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun f1s x) Hm1R real_1 Hf1sIx).
L175295
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun f1s x) (minus_SNo 1)).
L175297
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun f1s x) (andEL (Rle (minus_SNo 1) (apply_fun f1s x)) (Rle (apply_fun f1s x) 1) Hf1s_boundsI)).
L175298
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun f1s x)).
L175300
An exact proof term for the current goal is (RleE_nlt (apply_fun f1s x) 1 (andER (Rle (minus_SNo 1) (apply_fun f1s x)) (Rle (apply_fun f1s x) 1) Hf1s_boundsI)).
L175301
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun f1s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun f1s x).
L175303
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun f1s x) Hm1R Hm13R Hf1sRx HnotI1).
L175304
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun f1s x).
L175306
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun f1s x))) to the current goal.
L175307
Assume Hbad: Rlt (apply_fun f1s x) (minus_SNo 1).
L175307
Apply FalseE to the current goal.
L175308
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L175310
Assume Hok: Rlt (minus_SNo one_third) (apply_fun f1s x).
L175310
An exact proof term for the current goal is Hok.
L175311
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun f1s x) (minus_SNo one_third)).
L175313
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun f1s x) Hm13lt_fx).
L175313
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun f1s x) one_third ∨ Rlt 1 (apply_fun f1s x).
L175315
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun f1s x) H13R real_1 Hf1sRx HnotI3).
L175316
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun f1s x) one_third.
L175318
Apply (HnotI3_cases (Rlt (apply_fun f1s x) one_third)) to the current goal.
L175319
Assume Hok: Rlt (apply_fun f1s x) one_third.
L175319
An exact proof term for the current goal is Hok.
L175321
Assume Hbad: Rlt 1 (apply_fun f1s x).
L175321
Apply FalseE to the current goal.
L175322
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L175323
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun f1s x)).
L175325
An exact proof term for the current goal is (not_Rlt_sym (apply_fun f1s x) one_third Hfx_lt_13).
L175325
We prove the intermediate claim Hf1sI0: apply_fun f1s x ∈ I0.
L175327
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L175329
We prove the intermediate claim HxSep: apply_fun f1s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L175331
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun f1s x) Hf1sRx (andI (¬ (Rlt (apply_fun f1s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun f1s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L175335
rewrite the current goal using HI0_def (from left to right).
L175336
An exact proof term for the current goal is HxSep.
L175337
We prove the intermediate claim Hu1funI0: function_on u1 X I0.
L175339
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u1 Hu1contI0).
L175339
We prove the intermediate claim Hu1xI0: apply_fun u1 x ∈ I0.
L175341
An exact proof term for the current goal is (Hu1funI0 x HxX).
L175341
We prove the intermediate claim Hu1xR: apply_fun u1 x ∈ R.
L175343
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u1 x) Hu1xI0).
L175343
We prove the intermediate claim Hm_u1x_R: minus_SNo (apply_fun u1 x) ∈ R.
L175345
An exact proof term for the current goal is (real_minus_SNo (apply_fun u1 x) Hu1xR).
L175345
rewrite the current goal using (Hr1_apply x HxA) (from left to right).
L175346
We prove the intermediate claim Hr1xR: add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x)) ∈ R.
L175348
An exact proof term for the current goal is (real_add_SNo (apply_fun f1s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun f1s x) Hf1sI0) (minus_SNo (apply_fun u1 x)) Hm_u1x_R).
L175350
We prove the intermediate claim H23R: two_thirds ∈ R.
L175352
An exact proof term for the current goal is two_thirds_in_R.
L175352
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L175354
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L175354
We prove the intermediate claim Hf1s_bounds0: Rle (minus_SNo one_third) (apply_fun f1s x) ∧ Rle (apply_fun f1s x) one_third.
L175356
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun f1s x) Hm13R H13R Hf1sI0).
L175356
We prove the intermediate claim Hu1_bounds0: Rle (minus_SNo one_third) (apply_fun u1 x) ∧ Rle (apply_fun u1 x) one_third.
L175358
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u1 x) Hm13R H13R Hu1xI0).
L175358
We prove the intermediate claim Hm13_le_f1s: Rle (minus_SNo one_third) (apply_fun f1s x).
L175360
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun f1s x)) (Rle (apply_fun f1s x) one_third) Hf1s_bounds0).
L175360
We prove the intermediate claim Hf1s_le_13: Rle (apply_fun f1s x) one_third.
L175362
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun f1s x)) (Rle (apply_fun f1s x) one_third) Hf1s_bounds0).
L175362
We prove the intermediate claim Hm13_le_u1x: Rle (minus_SNo one_third) (apply_fun u1 x).
L175364
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u1 x)) (Rle (apply_fun u1 x) one_third) Hu1_bounds0).
L175364
We prove the intermediate claim Hu1x_le_13: Rle (apply_fun u1 x) one_third.
L175366
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u1 x)) (Rle (apply_fun u1 x) one_third) Hu1_bounds0).
L175366
We prove the intermediate claim Hm13_le_mu1: Rle (minus_SNo one_third) (minus_SNo (apply_fun u1 x)).
L175368
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u1 x) one_third Hu1x_le_13).
L175368
We prove the intermediate claim Hmu1_le_13: Rle (minus_SNo (apply_fun u1 x)) one_third.
L175370
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u1 x)) (minus_SNo (minus_SNo one_third)).
L175371
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u1 x) Hm13_le_u1x).
L175371
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L175372
An exact proof term for the current goal is Htmp.
L175373
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u1 x))).
L175376
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u1 x)) Hm13R Hm13R Hm_u1x_R Hm13_le_mu1).
L175377
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u1 x))) (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))).
L175380
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun f1s x) (minus_SNo (apply_fun u1 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun f1s x) Hf1sI0) Hm_u1x_R Hm13_le_f1s).
L175384
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))).
L175387
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u1 x))) (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))) Hlow1 Hlow2).
L175390
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))).
L175392
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L175392
An exact proof term for the current goal is Hlow_tmp.
L175393
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))) (add_SNo (apply_fun f1s x) one_third).
L175396
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun f1s x) (minus_SNo (apply_fun u1 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun f1s x) Hf1sI0) Hm_u1x_R H13R Hmu1_le_13).
L175398
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun f1s x) one_third) (add_SNo one_third one_third).
L175400
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun f1s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun f1s x) Hf1sI0) H13R H13R Hf1s_le_13).
L175402
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))) (add_SNo one_third one_third).
L175405
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))) (add_SNo (apply_fun f1s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L175408
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L175410
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))) two_thirds.
L175412
rewrite the current goal using Hdef23 (from left to right) at position 1.
L175412
An exact proof term for the current goal is Hup_tmp.
L175413
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))) Hm23R H23R Hr1xR Hlow Hup).
L175416
Set r1s to be the term compose_fun A r1 (div_const_fun den).
L175419
We prove the intermediate claim Hr1s_cont: continuous_map A Ta I Ti r1s.
(*** second-step scaled residual and second correction u2 ***)
L175421
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
(*** prove continuity into R, then restrict the range to I ***)
L175423
An exact proof term for the current goal is R_standard_topology_is_topology.
L175423
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L175425
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L175425
We prove the intermediate claim Hr1s_contR: continuous_map A Ta R R_standard_topology r1s.
L175427
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r1 (div_const_fun den) Hr1_cont Hdivcont).
L175428
We prove the intermediate claim Hr1s_I: ∀x : set, x ∈ A → apply_fun r1s x ∈ I.
L175430
Let x be given.
L175430
Assume HxA: x ∈ A.
L175430
We prove the intermediate claim Hr1xI2: apply_fun r1 x ∈ I2.
L175432
An exact proof term for the current goal is (Hr1_range x HxA).
L175432
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L175434
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L175434
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L175436
An exact proof term for the current goal is (real_minus_SNo den H23R).
L175436
We prove the intermediate claim Hr1xR: apply_fun r1 x ∈ R.
L175438
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r1 x) Hr1xI2).
L175438
We prove the intermediate claim Hr1xS: SNo (apply_fun r1 x).
L175440
An exact proof term for the current goal is (real_SNo (apply_fun r1 x) Hr1xR).
L175440
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r1 x) ∧ Rle (apply_fun r1 x) den.
L175442
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r1 x) HmdenR H23R Hr1xI2).
L175442
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r1 x).
L175444
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r1 x)) (Rle (apply_fun r1 x) den) Hbounds).
L175446
We prove the intermediate claim Hhi: Rle (apply_fun r1 x) den.
L175448
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r1 x)) (Rle (apply_fun r1 x) den) Hbounds).
L175450
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r1 x)).
L175452
An exact proof term for the current goal is (RleE_nlt (apply_fun r1 x) den Hhi).
L175452
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r1 x) (minus_SNo den)).
L175454
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r1 x) Hlo).
L175454
We prove the intermediate claim HyEq: apply_fun r1s x = div_SNo (apply_fun r1 x) den.
L175456
rewrite the current goal using (compose_fun_apply A r1 (div_const_fun den) x HxA) (from left to right).
L175456
rewrite the current goal using (div_const_fun_apply den (apply_fun r1 x) H23R Hr1xR) (from left to right).
Use reflexivity.
L175458
We prove the intermediate claim HyR: apply_fun r1s x ∈ R.
L175460
rewrite the current goal using HyEq (from left to right).
L175460
An exact proof term for the current goal is (real_div_SNo (apply_fun r1 x) Hr1xR den H23R).
L175461
We prove the intermediate claim HyS: SNo (apply_fun r1s x).
L175463
An exact proof term for the current goal is (real_SNo (apply_fun r1s x) HyR).
L175463
We prove the intermediate claim Hy_le_1: Rle (apply_fun r1s x) 1.
L175465
Apply (RleI (apply_fun r1s x) 1 HyR real_1) to the current goal.
L175465
We will prove ¬ (Rlt 1 (apply_fun r1s x)).
L175466
Assume H1lt: Rlt 1 (apply_fun r1s x).
L175467
We prove the intermediate claim H1lty: 1 < apply_fun r1s x.
L175469
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r1s x) H1lt).
L175469
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r1s x) den.
L175471
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r1s x) den SNo_1 HyS H23S H23pos H1lty).
L175471
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r1s x) den.
L175473
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L175473
An exact proof term for the current goal is HmulLt.
L175474
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r1s x) den = apply_fun r1 x.
L175476
rewrite the current goal using HyEq (from left to right) at position 1.
L175476
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r1 x) den Hr1xS H23S H23ne0).
L175477
We prove the intermediate claim Hden_lt_r1x: den < apply_fun r1 x.
L175479
rewrite the current goal using HmulEq (from right to left).
L175479
An exact proof term for the current goal is HmulLt'.
L175480
We prove the intermediate claim Hbad: Rlt den (apply_fun r1 x).
L175482
An exact proof term for the current goal is (RltI den (apply_fun r1 x) H23R Hr1xR Hden_lt_r1x).
L175482
An exact proof term for the current goal is (Hnlt_hi Hbad).
L175483
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r1s x).
L175485
Apply (RleI (minus_SNo 1) (apply_fun r1s x) Hm1R HyR) to the current goal.
L175485
We will prove ¬ (Rlt (apply_fun r1s x) (minus_SNo 1)).
L175486
Assume Hylt: Rlt (apply_fun r1s x) (minus_SNo 1).
L175487
We prove the intermediate claim Hylts: apply_fun r1s x < minus_SNo 1.
L175489
An exact proof term for the current goal is (RltE_lt (apply_fun r1s x) (minus_SNo 1) Hylt).
L175489
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r1s x) den < mul_SNo (minus_SNo 1) den.
L175491
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r1s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L175492
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r1s x) den = apply_fun r1 x.
L175494
rewrite the current goal using HyEq (from left to right) at position 1.
L175494
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r1 x) den Hr1xS H23S H23ne0).
L175495
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L175497
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L175497
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L175499
We prove the intermediate claim Hr1x_lt_mden: apply_fun r1 x < minus_SNo den.
L175501
rewrite the current goal using HmulEq (from right to left).
L175501
rewrite the current goal using HrhsEq (from right to left).
L175502
An exact proof term for the current goal is HmulLt.
L175503
We prove the intermediate claim Hbad: Rlt (apply_fun r1 x) (minus_SNo den).
L175505
An exact proof term for the current goal is (RltI (apply_fun r1 x) (minus_SNo den) Hr1xR HmdenR Hr1x_lt_mden).
L175505
An exact proof term for the current goal is (Hnlt_lo Hbad).
L175506
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r1s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L175507
We will prove continuous_map A Ta I Ti r1s.
L175508
rewrite the current goal using HTiEq (from left to right).
L175509
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r1s I Hr1s_contR HIcR Hr1s_I).
L175510
We prove the intermediate claim Hex_u2: ∃u2 : set, continuous_map X Tx I0 T0 u2 ∧ (∀x : set, x ∈ preimage_of A r1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u2 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r1s ((closed_interval one_third 1) ∩ I) → apply_fun u2 x = one_third).
L175518
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r1s Hnorm HA Hr1s_cont).
L175518
Apply Hex_u2 to the current goal.
L175519
Let u2 be given.
L175520
Assume Hu2.
L175520
We prove the intermediate claim Hu2AB: continuous_map X Tx I0 T0 u2 ∧ (∀x : set, x ∈ preimage_of A r1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u2 x = minus_SNo one_third).
L175525
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u2 ∧ (∀x : set, x ∈ preimage_of A r1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u2 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r1s ((closed_interval one_third 1) ∩ I) → apply_fun u2 x = one_third) Hu2).
L175531
We prove the intermediate claim Hu2contI0: continuous_map X Tx I0 T0 u2.
L175533
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u2) (∀x : set, x ∈ preimage_of A r1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u2 x = minus_SNo one_third) Hu2AB).
L175537
We prove the intermediate claim Hu2contR: continuous_map X Tx R R_standard_topology u2.
L175539
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L175540
We prove the intermediate claim HI0subR: I0 ⊆ R.
L175542
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L175542
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u2 Hu2contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L175550
Set den2 to be the term mul_SNo den den.
L175553
We prove the intermediate claim Hden2R: den2 ∈ R.
(*** a second partial sum g2 = g1 + (2/3)^2 u2 is continuous ***)
L175555
An exact proof term for the current goal is (real_mul_SNo den HdenR den HdenR).
L175555
We prove the intermediate claim Hden2pos: 0 < den2.
L175557
An exact proof term for the current goal is (mul_SNo_pos_pos den den H23S H23S HdenPos HdenPos).
L175557
Set u2s to be the term compose_fun X u2 (mul_const_fun den2).
L175558
We prove the intermediate claim Hu2s_cont: continuous_map X Tx R R_standard_topology u2s.
L175560
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u2 den2 HTx Hu2contR Hden2R Hden2pos).
L175560
Set g2 to be the term compose_fun X (pair_map X g1 u2s) add_fun_R.
L175561
We prove the intermediate claim Hg2cont: continuous_map X Tx R R_standard_topology g2.
L175563
An exact proof term for the current goal is (add_two_continuous_R X Tx g1 u2s HTx Hg1cont Hu2s_cont).
L175563
We prove the intermediate claim Hu2contA: continuous_map A Ta R R_standard_topology u2.
(*** third correction step scaffold: residual r2 on A, scaling r2s, and u3 ***)
L175567
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u2 A HTx HAsubX Hu2contR).
L175567
Set u2neg to be the term compose_fun A u2 neg_fun.
L175568
We prove the intermediate claim Hu2neg_cont: continuous_map A Ta R R_standard_topology u2neg.
L175570
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u2 neg_fun Hu2contA Hnegcont).
L175571
We prove the intermediate claim Hr1s_contR: continuous_map A Ta R R_standard_topology r1s.
L175573
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r1s Hr1s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L175574
Set r2 to be the term compose_fun A (pair_map A r1s u2neg) add_fun_R.
L175575
We prove the intermediate claim Hr2_cont: continuous_map A Ta R R_standard_topology r2.
L175577
An exact proof term for the current goal is (add_two_continuous_R A Ta r1s u2neg HTa Hr1s_contR Hu2neg_cont).
L175577
We prove the intermediate claim Hr2_apply: ∀x : set, x ∈ A → apply_fun r2 x = add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x)).
L175580
Let x be given.
L175580
Assume HxA: x ∈ A.
L175580
We prove the intermediate claim Hpimg: apply_fun (pair_map A r1s u2neg) x ∈ setprod R R.
L175582
rewrite the current goal using (pair_map_apply A R R r1s u2neg x HxA) (from left to right).
L175582
We prove the intermediate claim Hr1sRx: apply_fun r1s x ∈ R.
L175584
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology r1s Hr1s_contR x HxA).
L175584
We prove the intermediate claim Hu2negRx: apply_fun u2neg x ∈ R.
L175586
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u2neg Hu2neg_cont x HxA).
L175586
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r1s x) (apply_fun u2neg x) Hr1sRx Hu2negRx).
L175587
rewrite the current goal using (compose_fun_apply A (pair_map A r1s u2neg) add_fun_R x HxA) (from left to right).
L175588
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r1s u2neg) x) Hpimg) (from left to right) at position 1.
L175589
rewrite the current goal using (pair_map_apply A R R r1s u2neg x HxA) (from left to right).
L175590
rewrite the current goal using (tuple_2_0_eq (apply_fun r1s x) (apply_fun u2neg x)) (from left to right).
L175591
rewrite the current goal using (tuple_2_1_eq (apply_fun r1s x) (apply_fun u2neg x)) (from left to right).
L175592
rewrite the current goal using (compose_fun_apply A u2 neg_fun x HxA) (from left to right) at position 1.
L175593
We prove the intermediate claim Hu2Rx: apply_fun u2 x ∈ R.
L175595
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u2 Hu2contA x HxA).
L175595
rewrite the current goal using (neg_fun_apply (apply_fun u2 x) Hu2Rx) (from left to right) at position 1.
Use reflexivity.
L175597
We prove the intermediate claim Hr2_range: ∀x : set, x ∈ A → apply_fun r2 x ∈ I2.
L175599
We prove the intermediate claim Hu2_on_B2: ∀x : set, x ∈ preimage_of A r1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u2 x = minus_SNo one_third.
(*** show the next residual stays within [-2/3,2/3] on A ***)
L175603
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u2) (∀x : set, x ∈ preimage_of A r1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u2 x = minus_SNo one_third) Hu2AB).
L175607
We prove the intermediate claim Hu2_on_C2: ∀x : set, x ∈ preimage_of A r1s ((closed_interval one_third 1) ∩ I) → apply_fun u2 x = one_third.
L175611
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u2 ∧ (∀x : set, x ∈ preimage_of A r1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u2 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r1s ((closed_interval one_third 1) ∩ I) → apply_fun u2 x = one_third) Hu2).
L175617
Let x be given.
L175618
Assume HxA: x ∈ A.
L175618
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L175619
Set I3 to be the term closed_interval one_third 1.
L175620
Set B2 to be the term preimage_of A r1s (I1 ∩ I).
L175621
Set C2 to be the term preimage_of A r1s (I3 ∩ I).
L175622
We prove the intermediate claim Hr1sIx: apply_fun r1s x ∈ I.
L175624
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r1s Hr1s_cont x HxA).
L175624
We prove the intermediate claim HB2_cases: x ∈ B2 ∨ ¬ (x ∈ B2).
L175626
An exact proof term for the current goal is (xm (x ∈ B2)).
L175626
Apply (HB2_cases (apply_fun r2 x ∈ I2)) to the current goal.
L175628
Assume HxB2: x ∈ B2.
L175628
We prove the intermediate claim Hu2eq: apply_fun u2 x = minus_SNo one_third.
L175630
An exact proof term for the current goal is (Hu2_on_B2 x HxB2).
L175630
We prove the intermediate claim Hr2eq: apply_fun r2 x = add_SNo (apply_fun r1s x) one_third.
L175632
rewrite the current goal using (Hr2_apply x HxA) (from left to right).
L175632
rewrite the current goal using Hu2eq (from left to right) at position 1.
L175633
We prove the intermediate claim H13R: one_third ∈ R.
L175635
An exact proof term for the current goal is one_third_in_R.
L175635
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L175637
rewrite the current goal using Hr2eq (from left to right).
L175638
We prove the intermediate claim Hr1sI1I: apply_fun r1s x ∈ I1 ∩ I.
L175640
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r1s x0 ∈ I1 ∩ I) x HxB2).
L175640
We prove the intermediate claim Hr1sI1: apply_fun r1s x ∈ I1.
L175642
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r1s x) Hr1sI1I).
L175642
We prove the intermediate claim H13R: one_third ∈ R.
L175644
An exact proof term for the current goal is one_third_in_R.
L175644
We prove the intermediate claim H23R: two_thirds ∈ R.
L175646
An exact proof term for the current goal is two_thirds_in_R.
L175646
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L175648
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L175648
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L175650
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L175650
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L175652
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L175652
We prove the intermediate claim Hr1sRx: apply_fun r1s x ∈ R.
L175654
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r1s x) Hr1sI1).
L175654
We prove the intermediate claim Hr1s_bounds: Rle (minus_SNo 1) (apply_fun r1s x) ∧ Rle (apply_fun r1s x) (minus_SNo one_third).
L175656
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r1s x) Hm1R Hm13R Hr1sI1).
L175657
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r1s x).
L175659
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r1s x)) (Rle (apply_fun r1s x) (minus_SNo one_third)) Hr1s_bounds).
L175660
We prove the intermediate claim HhiI1: Rle (apply_fun r1s x) (minus_SNo one_third).
L175662
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r1s x)) (Rle (apply_fun r1s x) (minus_SNo one_third)) Hr1s_bounds).
L175663
We prove the intermediate claim Hr2Rx: add_SNo (apply_fun r1s x) one_third ∈ R.
L175665
An exact proof term for the current goal is (real_add_SNo (apply_fun r1s x) Hr1sRx one_third H13R).
L175665
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r1s x) one_third).
L175667
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r1s x) one_third Hm1R Hr1sRx H13R Hm1le).
L175667
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r1s x) one_third).
L175669
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L175669
An exact proof term for the current goal is Hlow_tmp.
L175670
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r1s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L175672
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r1s x) (minus_SNo one_third) one_third Hr1sRx Hm13R H13R HhiI1).
L175672
We prove the intermediate claim H13S: SNo one_third.
L175674
An exact proof term for the current goal is (real_SNo one_third H13R).
L175674
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r1s x) one_third) 0.
L175676
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L175676
An exact proof term for the current goal is Hup0_tmp.
L175677
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L175679
An exact proof term for the current goal is Rle_0_two_thirds.
L175679
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r1s x) one_third) two_thirds.
L175681
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r1s x) one_third) 0 two_thirds Hup0 H0le23).
L175681
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r1s x) one_third) Hm23R H23R Hr2Rx Hlow Hup).
L175685
Assume HxnotB2: ¬ (x ∈ B2).
L175685
We prove the intermediate claim HC2_cases: x ∈ C2 ∨ ¬ (x ∈ C2).
L175687
An exact proof term for the current goal is (xm (x ∈ C2)).
L175687
Apply (HC2_cases (apply_fun r2 x ∈ I2)) to the current goal.
L175689
Assume HxC2: x ∈ C2.
L175689
We prove the intermediate claim Hu2eq: apply_fun u2 x = one_third.
L175691
An exact proof term for the current goal is (Hu2_on_C2 x HxC2).
L175691
We prove the intermediate claim Hr2eq: apply_fun r2 x = add_SNo (apply_fun r1s x) (minus_SNo one_third).
L175693
rewrite the current goal using (Hr2_apply x HxA) (from left to right).
L175693
rewrite the current goal using Hu2eq (from left to right) at position 1.
Use reflexivity.
L175695
rewrite the current goal using Hr2eq (from left to right).
L175696
We prove the intermediate claim Hr1sI3I: apply_fun r1s x ∈ I3 ∩ I.
L175698
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r1s x0 ∈ I3 ∩ I) x HxC2).
L175698
We prove the intermediate claim Hr1sI3: apply_fun r1s x ∈ I3.
L175700
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r1s x) Hr1sI3I).
L175700
We prove the intermediate claim H13R: one_third ∈ R.
L175702
An exact proof term for the current goal is one_third_in_R.
L175702
We prove the intermediate claim H23R: two_thirds ∈ R.
L175704
An exact proof term for the current goal is two_thirds_in_R.
L175704
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L175706
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L175706
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L175708
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L175708
We prove the intermediate claim Hr1sRx: apply_fun r1s x ∈ R.
L175710
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r1s x) Hr1sI3).
L175710
We prove the intermediate claim Hr1s_bounds: Rle one_third (apply_fun r1s x) ∧ Rle (apply_fun r1s x) 1.
L175712
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r1s x) H13R real_1 Hr1sI3).
L175712
We prove the intermediate claim HloI3: Rle one_third (apply_fun r1s x).
L175714
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r1s x)) (Rle (apply_fun r1s x) 1) Hr1s_bounds).
L175714
We prove the intermediate claim HhiI3: Rle (apply_fun r1s x) 1.
L175716
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r1s x)) (Rle (apply_fun r1s x) 1) Hr1s_bounds).
L175716
We prove the intermediate claim Hr2Rx: add_SNo (apply_fun r1s x) (minus_SNo one_third) ∈ R.
L175718
An exact proof term for the current goal is (real_add_SNo (apply_fun r1s x) Hr1sRx (minus_SNo one_third) Hm13R).
L175718
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r1s x) (minus_SNo one_third)).
L175721
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r1s x) (minus_SNo one_third) H13R Hr1sRx Hm13R HloI3).
L175722
We prove the intermediate claim H13S: SNo one_third.
L175724
An exact proof term for the current goal is (real_SNo one_third H13R).
L175724
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r1s x) (minus_SNo one_third)).
L175726
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L175726
An exact proof term for the current goal is H0le_tmp.
L175727
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L175729
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L175729
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r1s x) (minus_SNo one_third)).
L175731
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r1s x) (minus_SNo one_third)) Hm23le0 H0le).
L175733
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r1s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L175736
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r1s x) 1 (minus_SNo one_third) Hr1sRx real_1 Hm13R HhiI3).
L175737
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L175739
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L175739
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L175740
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r1s x) (minus_SNo one_third)) two_thirds.
L175742
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r1s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L175745
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r1s x) (minus_SNo one_third)) Hm23R H23R Hr2Rx Hlow Hup).
L175749
Assume HxnotC2: ¬ (x ∈ C2).
L175749
We prove the intermediate claim HxX: x ∈ X.
L175751
An exact proof term for the current goal is (HAsubX x HxA).
L175751
We prove the intermediate claim HnotI1: ¬ (apply_fun r1s x ∈ I1).
L175753
Assume Hr1sI1': apply_fun r1s x ∈ I1.
L175753
We prove the intermediate claim Hr1sI1I: apply_fun r1s x ∈ I1 ∩ I.
L175755
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r1s x) Hr1sI1' Hr1sIx).
L175755
We prove the intermediate claim HxB2': x ∈ B2.
L175757
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r1s x0 ∈ I1 ∩ I) x HxA Hr1sI1I).
L175757
Apply FalseE to the current goal.
L175758
An exact proof term for the current goal is (HxnotB2 HxB2').
L175759
We prove the intermediate claim HnotI3: ¬ (apply_fun r1s x ∈ I3).
L175761
Assume Hr1sI3': apply_fun r1s x ∈ I3.
L175761
We prove the intermediate claim Hr1sI3I: apply_fun r1s x ∈ I3 ∩ I.
L175763
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r1s x) Hr1sI3' Hr1sIx).
L175763
We prove the intermediate claim HxC2': x ∈ C2.
L175765
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r1s x0 ∈ I3 ∩ I) x HxA Hr1sI3I).
L175765
Apply FalseE to the current goal.
L175766
An exact proof term for the current goal is (HxnotC2 HxC2').
L175767
We prove the intermediate claim Hr1sRx: apply_fun r1s x ∈ R.
L175769
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r1s x) Hr1sIx).
L175769
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L175771
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L175771
We prove the intermediate claim H13R: one_third ∈ R.
L175773
An exact proof term for the current goal is one_third_in_R.
L175773
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L175775
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L175775
We prove the intermediate claim Hr1s_boundsI: Rle (minus_SNo 1) (apply_fun r1s x) ∧ Rle (apply_fun r1s x) 1.
L175777
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r1s x) Hm1R real_1 Hr1sIx).
L175777
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r1s x) (minus_SNo 1)).
L175779
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r1s x) (andEL (Rle (minus_SNo 1) (apply_fun r1s x)) (Rle (apply_fun r1s x) 1) Hr1s_boundsI)).
L175780
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r1s x)).
L175782
An exact proof term for the current goal is (RleE_nlt (apply_fun r1s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r1s x)) (Rle (apply_fun r1s x) 1) Hr1s_boundsI)).
L175783
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r1s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r1s x).
L175785
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r1s x) Hm1R Hm13R Hr1sRx HnotI1).
L175786
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r1s x).
L175788
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r1s x))) to the current goal.
L175789
Assume Hbad: Rlt (apply_fun r1s x) (minus_SNo 1).
L175789
Apply FalseE to the current goal.
L175790
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L175792
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r1s x).
L175792
An exact proof term for the current goal is Hok.
L175793
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r1s x) (minus_SNo one_third)).
L175795
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r1s x) Hm13lt_fx).
L175795
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r1s x) one_third ∨ Rlt 1 (apply_fun r1s x).
L175797
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r1s x) H13R real_1 Hr1sRx HnotI3).
L175798
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r1s x) one_third.
L175800
Apply (HnotI3_cases (Rlt (apply_fun r1s x) one_third)) to the current goal.
L175801
Assume Hok: Rlt (apply_fun r1s x) one_third.
L175801
An exact proof term for the current goal is Hok.
L175803
Assume Hbad: Rlt 1 (apply_fun r1s x).
L175803
Apply FalseE to the current goal.
L175804
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L175805
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r1s x)).
L175807
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r1s x) one_third Hfx_lt_13).
L175807
We prove the intermediate claim Hr1sI0: apply_fun r1s x ∈ I0.
L175809
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L175811
We prove the intermediate claim HxSep: apply_fun r1s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L175813
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r1s x) Hr1sRx (andI (¬ (Rlt (apply_fun r1s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r1s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L175817
rewrite the current goal using HI0_def (from left to right).
L175818
An exact proof term for the current goal is HxSep.
L175819
We prove the intermediate claim Hu2funI0: function_on u2 X I0.
L175821
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u2 Hu2contI0).
L175821
We prove the intermediate claim Hu2xI0: apply_fun u2 x ∈ I0.
L175823
An exact proof term for the current goal is (Hu2funI0 x HxX).
L175823
We prove the intermediate claim Hu2xR: apply_fun u2 x ∈ R.
L175825
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u2 x) Hu2xI0).
L175825
We prove the intermediate claim Hm_u2x_R: minus_SNo (apply_fun u2 x) ∈ R.
L175827
An exact proof term for the current goal is (real_minus_SNo (apply_fun u2 x) Hu2xR).
L175827
rewrite the current goal using (Hr2_apply x HxA) (from left to right).
L175828
We prove the intermediate claim Hr2xR: add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x)) ∈ R.
L175830
An exact proof term for the current goal is (real_add_SNo (apply_fun r1s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r1s x) Hr1sI0) (minus_SNo (apply_fun u2 x)) Hm_u2x_R).
L175832
We prove the intermediate claim H23R: two_thirds ∈ R.
L175834
An exact proof term for the current goal is two_thirds_in_R.
L175834
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L175836
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L175836
We prove the intermediate claim Hr1s_bounds0: Rle (minus_SNo one_third) (apply_fun r1s x) ∧ Rle (apply_fun r1s x) one_third.
L175838
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r1s x) Hm13R H13R Hr1sI0).
L175838
We prove the intermediate claim Hu2_bounds0: Rle (minus_SNo one_third) (apply_fun u2 x) ∧ Rle (apply_fun u2 x) one_third.
L175840
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u2 x) Hm13R H13R Hu2xI0).
L175840
We prove the intermediate claim Hm13_le_r1s: Rle (minus_SNo one_third) (apply_fun r1s x).
L175842
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r1s x)) (Rle (apply_fun r1s x) one_third) Hr1s_bounds0).
L175842
We prove the intermediate claim Hr1s_le_13: Rle (apply_fun r1s x) one_third.
L175844
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r1s x)) (Rle (apply_fun r1s x) one_third) Hr1s_bounds0).
L175844
We prove the intermediate claim Hm13_le_u2x: Rle (minus_SNo one_third) (apply_fun u2 x).
L175846
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u2 x)) (Rle (apply_fun u2 x) one_third) Hu2_bounds0).
L175846
We prove the intermediate claim Hu2x_le_13: Rle (apply_fun u2 x) one_third.
L175848
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u2 x)) (Rle (apply_fun u2 x) one_third) Hu2_bounds0).
L175848
We prove the intermediate claim Hm13_le_mu2: Rle (minus_SNo one_third) (minus_SNo (apply_fun u2 x)).
L175850
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u2 x) one_third Hu2x_le_13).
L175850
We prove the intermediate claim Hmu2_le_13: Rle (minus_SNo (apply_fun u2 x)) one_third.
L175852
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u2 x)) (minus_SNo (minus_SNo one_third)).
L175853
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u2 x) Hm13_le_u2x).
L175853
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L175854
An exact proof term for the current goal is Htmp.
L175855
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u2 x))).
L175858
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u2 x)) Hm13R Hm13R Hm_u2x_R Hm13_le_mu2).
L175859
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u2 x))) (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))).
L175862
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r1s x) (minus_SNo (apply_fun u2 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r1s x) Hr1sI0) Hm_u2x_R Hm13_le_r1s).
L175866
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))).
L175869
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u2 x))) (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))) Hlow1 Hlow2).
L175872
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))).
L175874
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L175874
An exact proof term for the current goal is Hlow_tmp.
L175875
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))) (add_SNo (apply_fun r1s x) one_third).
L175878
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r1s x) (minus_SNo (apply_fun u2 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r1s x) Hr1sI0) Hm_u2x_R H13R Hmu2_le_13).
L175880
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r1s x) one_third) (add_SNo one_third one_third).
L175883
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r1s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r1s x) Hr1sI0) H13R H13R Hr1s_le_13).
L175885
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))) (add_SNo one_third one_third).
L175888
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))) (add_SNo (apply_fun r1s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L175891
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))) two_thirds.
L175893
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L175894
rewrite the current goal using Hdef23 (from left to right).
L175895
An exact proof term for the current goal is Hup_tmp.
L175896
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))) Hm23R H23R Hr2xR Hlow Hup).
L175899
Set r2s to be the term compose_fun A r2 (div_const_fun den).
L175900
We prove the intermediate claim Hr2s_cont: continuous_map A Ta I Ti r2s.
L175902
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
L175903
An exact proof term for the current goal is R_standard_topology_is_topology.
L175903
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L175905
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L175905
We prove the intermediate claim Hr2s_contR: continuous_map A Ta R R_standard_topology r2s.
L175907
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r2 (div_const_fun den) Hr2_cont Hdivcont).
L175908
We prove the intermediate claim Hr2s_I: ∀x : set, x ∈ A → apply_fun r2s x ∈ I.
L175910
Let x be given.
L175910
Assume HxA: x ∈ A.
L175910
We prove the intermediate claim Hr2xI2: apply_fun r2 x ∈ I2.
L175912
An exact proof term for the current goal is (Hr2_range x HxA).
L175912
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L175914
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L175914
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L175916
An exact proof term for the current goal is (real_minus_SNo den H23R).
L175916
We prove the intermediate claim Hr2xR: apply_fun r2 x ∈ R.
L175918
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r2 x) Hr2xI2).
L175918
We prove the intermediate claim Hr2xS: SNo (apply_fun r2 x).
L175920
An exact proof term for the current goal is (real_SNo (apply_fun r2 x) Hr2xR).
L175920
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r2 x) ∧ Rle (apply_fun r2 x) den.
L175922
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r2 x) HmdenR H23R Hr2xI2).
L175922
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r2 x).
L175924
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r2 x)) (Rle (apply_fun r2 x) den) Hbounds).
L175926
We prove the intermediate claim Hhi: Rle (apply_fun r2 x) den.
L175928
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r2 x)) (Rle (apply_fun r2 x) den) Hbounds).
L175930
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r2 x)).
L175932
An exact proof term for the current goal is (RleE_nlt (apply_fun r2 x) den Hhi).
L175932
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r2 x) (minus_SNo den)).
L175934
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r2 x) Hlo).
L175934
We prove the intermediate claim HyEq: apply_fun r2s x = div_SNo (apply_fun r2 x) den.
L175936
rewrite the current goal using (compose_fun_apply A r2 (div_const_fun den) x HxA) (from left to right).
L175936
rewrite the current goal using (div_const_fun_apply den (apply_fun r2 x) H23R Hr2xR) (from left to right).
Use reflexivity.
L175938
We prove the intermediate claim HyR: apply_fun r2s x ∈ R.
L175940
rewrite the current goal using HyEq (from left to right).
L175940
An exact proof term for the current goal is (real_div_SNo (apply_fun r2 x) Hr2xR den H23R).
L175941
We prove the intermediate claim HyS: SNo (apply_fun r2s x).
L175943
An exact proof term for the current goal is (real_SNo (apply_fun r2s x) HyR).
L175943
We prove the intermediate claim Hy_le_1: Rle (apply_fun r2s x) 1.
L175945
Apply (RleI (apply_fun r2s x) 1 HyR real_1) to the current goal.
L175945
We will prove ¬ (Rlt 1 (apply_fun r2s x)).
L175946
Assume H1lt: Rlt 1 (apply_fun r2s x).
L175947
We prove the intermediate claim H1lty: 1 < apply_fun r2s x.
L175949
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r2s x) H1lt).
L175949
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r2s x) den.
L175951
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r2s x) den SNo_1 HyS H23S H23pos H1lty).
L175951
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r2s x) den.
L175953
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L175953
An exact proof term for the current goal is HmulLt.
L175954
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r2s x) den = apply_fun r2 x.
L175956
rewrite the current goal using HyEq (from left to right) at position 1.
L175956
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r2 x) den Hr2xS H23S H23ne0).
L175957
We prove the intermediate claim Hden_lt_r2x: den < apply_fun r2 x.
L175959
rewrite the current goal using HmulEq (from right to left).
L175959
An exact proof term for the current goal is HmulLt'.
L175960
We prove the intermediate claim Hbad: Rlt den (apply_fun r2 x).
L175962
An exact proof term for the current goal is (RltI den (apply_fun r2 x) H23R Hr2xR Hden_lt_r2x).
L175962
An exact proof term for the current goal is (Hnlt_hi Hbad).
L175963
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r2s x).
L175965
Apply (RleI (minus_SNo 1) (apply_fun r2s x) Hm1R HyR) to the current goal.
L175965
We will prove ¬ (Rlt (apply_fun r2s x) (minus_SNo 1)).
L175966
Assume Hylt: Rlt (apply_fun r2s x) (minus_SNo 1).
L175967
We prove the intermediate claim Hylts: apply_fun r2s x < minus_SNo 1.
L175969
An exact proof term for the current goal is (RltE_lt (apply_fun r2s x) (minus_SNo 1) Hylt).
L175969
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r2s x) den < mul_SNo (minus_SNo 1) den.
L175971
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r2s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L175972
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r2s x) den = apply_fun r2 x.
L175974
rewrite the current goal using HyEq (from left to right) at position 1.
L175974
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r2 x) den Hr2xS H23S H23ne0).
L175975
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L175977
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L175977
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L175979
We prove the intermediate claim Hr2x_lt_mden: apply_fun r2 x < minus_SNo den.
L175981
rewrite the current goal using HmulEq (from right to left).
L175981
rewrite the current goal using HrhsEq (from right to left).
L175982
An exact proof term for the current goal is HmulLt.
L175983
We prove the intermediate claim Hbad: Rlt (apply_fun r2 x) (minus_SNo den).
L175985
An exact proof term for the current goal is (RltI (apply_fun r2 x) (minus_SNo den) Hr2xR HmdenR Hr2x_lt_mden).
L175985
An exact proof term for the current goal is (Hnlt_lo Hbad).
L175986
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r2s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L175988
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r2s I Hr2s_contR HIcR Hr2s_I).
L175989
We prove the intermediate claim Hex_u3: ∃u3 : set, continuous_map X Tx I0 T0 u3 ∧ (∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r2s ((closed_interval one_third 1) ∩ I) → apply_fun u3 x = one_third).
L175997
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r2s Hnorm HA Hr2s_cont).
L175997
Apply Hex_u3 to the current goal.
L175998
Let u3 be given.
L175999
Assume Hu3.
L175999
We prove the intermediate claim Hu3contI0: continuous_map X Tx I0 T0 u3.
L176001
We prove the intermediate claim Hu3AB: continuous_map X Tx I0 T0 u3 ∧ (∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third).
L176005
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u3 ∧ (∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r2s ((closed_interval one_third 1) ∩ I) → apply_fun u3 x = one_third) Hu3).
L176011
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u3) (∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third) Hu3AB).
L176016
We prove the intermediate claim Hu3contR: continuous_map X Tx R R_standard_topology u3.
L176018
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L176019
We prove the intermediate claim HI0subR: I0 ⊆ R.
L176021
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L176021
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u3 Hu3contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L176029
Set den3 to be the term mul_SNo den2 den.
L176030
We prove the intermediate claim Hden3R: den3 ∈ R.
L176032
An exact proof term for the current goal is (real_mul_SNo den2 Hden2R den HdenR).
L176032
We prove the intermediate claim Hden3pos: 0 < den3.
L176034
We prove the intermediate claim Hden2S: SNo den2.
L176035
An exact proof term for the current goal is (real_SNo den2 Hden2R).
L176035
An exact proof term for the current goal is (mul_SNo_pos_pos den2 den Hden2S H23S Hden2pos HdenPos).
L176036
Set u3s to be the term compose_fun X u3 (mul_const_fun den3).
L176037
We prove the intermediate claim Hu3s_cont: continuous_map X Tx R R_standard_topology u3s.
L176039
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u3 den3 HTx Hu3contR Hden3R Hden3pos).
L176039
Set g3 to be the term compose_fun X (pair_map X g2 u3s) add_fun_R.
L176040
We prove the intermediate claim Hg3cont: continuous_map X Tx R R_standard_topology g3.
L176042
An exact proof term for the current goal is (add_two_continuous_R X Tx g2 u3s HTx Hg2cont Hu3s_cont).
L176042
We prove the intermediate claim Hu3contA: continuous_map A Ta R R_standard_topology u3.
(*** fourth correction step scaffold: residual r3 on A, scaling r3s, and u4 ***)
L176046
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u3 A HTx HAsubX Hu3contR).
L176046
Set u3neg to be the term compose_fun A u3 neg_fun.
L176047
We prove the intermediate claim Hu3neg_cont: continuous_map A Ta R R_standard_topology u3neg.
L176049
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u3 neg_fun Hu3contA Hnegcont).
L176050
We prove the intermediate claim Hr2s_contR: continuous_map A Ta R R_standard_topology r2s.
L176052
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r2s Hr2s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L176053
Set r3 to be the term compose_fun A (pair_map A r2s u3neg) add_fun_R.
L176054
We prove the intermediate claim Hr3_cont: continuous_map A Ta R R_standard_topology r3.
L176056
An exact proof term for the current goal is (add_two_continuous_R A Ta r2s u3neg HTa Hr2s_contR Hu3neg_cont).
L176056
We prove the intermediate claim Hr3_apply: ∀x : set, x ∈ A → apply_fun r3 x = add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x)).
L176059
Let x be given.
L176059
Assume HxA: x ∈ A.
L176059
We prove the intermediate claim Hpimg: apply_fun (pair_map A r2s u3neg) x ∈ setprod R R.
L176061
rewrite the current goal using (pair_map_apply A R R r2s u3neg x HxA) (from left to right).
L176061
We prove the intermediate claim Hr2sRx: apply_fun r2s x ∈ R.
L176063
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology r2s Hr2s_contR x HxA).
L176063
We prove the intermediate claim Hu3negRx: apply_fun u3neg x ∈ R.
L176065
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u3neg Hu3neg_cont x HxA).
L176065
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r2s x) (apply_fun u3neg x) Hr2sRx Hu3negRx).
L176066
rewrite the current goal using (compose_fun_apply A (pair_map A r2s u3neg) add_fun_R x HxA) (from left to right).
L176067
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r2s u3neg) x) Hpimg) (from left to right) at position 1.
L176068
rewrite the current goal using (pair_map_apply A R R r2s u3neg x HxA) (from left to right).
L176069
rewrite the current goal using (tuple_2_0_eq (apply_fun r2s x) (apply_fun u3neg x)) (from left to right).
L176070
rewrite the current goal using (tuple_2_1_eq (apply_fun r2s x) (apply_fun u3neg x)) (from left to right).
L176071
rewrite the current goal using (compose_fun_apply A u3 neg_fun x HxA) (from left to right) at position 1.
L176072
We prove the intermediate claim Hu3Rx: apply_fun u3 x ∈ R.
L176074
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u3 Hu3contA x HxA).
L176074
rewrite the current goal using (neg_fun_apply (apply_fun u3 x) Hu3Rx) (from left to right) at position 1.
Use reflexivity.
L176076
We prove the intermediate claim Hr3_range: ∀x : set, x ∈ A → apply_fun r3 x ∈ I2.
L176078
We prove the intermediate claim Hu3AB: continuous_map X Tx I0 T0 u3 ∧ (∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third).
(*** show the next residual stays within [-2/3,2/3] on A ***)
L176083
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u3 ∧ (∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r2s ((closed_interval one_third 1) ∩ I) → apply_fun u3 x = one_third) Hu3).
L176089
We prove the intermediate claim Hu3_on_B3: ∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third.
L176093
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u3) (∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third) Hu3AB).
L176097
We prove the intermediate claim Hu3_on_C3: ∀x : set, x ∈ preimage_of A r2s ((closed_interval one_third 1) ∩ I) → apply_fun u3 x = one_third.
L176101
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u3 ∧ (∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r2s ((closed_interval one_third 1) ∩ I) → apply_fun u3 x = one_third) Hu3).
L176107
Let x be given.
L176108
Assume HxA: x ∈ A.
L176108
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L176109
Set I3 to be the term closed_interval one_third 1.
L176110
Set B3 to be the term preimage_of A r2s (I1 ∩ I).
L176111
Set C3 to be the term preimage_of A r2s (I3 ∩ I).
L176112
We prove the intermediate claim Hr2sIx: apply_fun r2s x ∈ I.
L176114
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r2s Hr2s_cont x HxA).
L176114
We prove the intermediate claim HB3_cases: x ∈ B3 ∨ ¬ (x ∈ B3).
L176116
An exact proof term for the current goal is (xm (x ∈ B3)).
L176116
Apply (HB3_cases (apply_fun r3 x ∈ I2)) to the current goal.
L176118
Assume HxB3: x ∈ B3.
L176118
We prove the intermediate claim Hu3eq: apply_fun u3 x = minus_SNo one_third.
L176120
An exact proof term for the current goal is (Hu3_on_B3 x HxB3).
L176120
We prove the intermediate claim Hr3eq: apply_fun r3 x = add_SNo (apply_fun r2s x) one_third.
L176122
rewrite the current goal using (Hr3_apply x HxA) (from left to right).
L176122
rewrite the current goal using Hu3eq (from left to right) at position 1.
L176123
We prove the intermediate claim H13R: one_third ∈ R.
L176125
An exact proof term for the current goal is one_third_in_R.
L176125
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L176127
rewrite the current goal using Hr3eq (from left to right).
L176128
We prove the intermediate claim Hr2sI1I: apply_fun r2s x ∈ I1 ∩ I.
L176130
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r2s x0 ∈ I1 ∩ I) x HxB3).
L176130
We prove the intermediate claim Hr2sI1: apply_fun r2s x ∈ I1.
L176132
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r2s x) Hr2sI1I).
L176132
We prove the intermediate claim H13R: one_third ∈ R.
L176134
An exact proof term for the current goal is one_third_in_R.
L176134
We prove the intermediate claim H23R: two_thirds ∈ R.
L176136
An exact proof term for the current goal is two_thirds_in_R.
L176136
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L176138
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L176138
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L176140
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L176140
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L176142
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L176142
We prove the intermediate claim Hr2sRx: apply_fun r2s x ∈ R.
L176144
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r2s x) Hr2sI1).
L176144
We prove the intermediate claim Hr2s_bounds: Rle (minus_SNo 1) (apply_fun r2s x) ∧ Rle (apply_fun r2s x) (minus_SNo one_third).
L176146
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r2s x) Hm1R Hm13R Hr2sI1).
L176147
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r2s x).
L176149
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r2s x)) (Rle (apply_fun r2s x) (minus_SNo one_third)) Hr2s_bounds).
L176150
We prove the intermediate claim HhiI1: Rle (apply_fun r2s x) (minus_SNo one_third).
L176152
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r2s x)) (Rle (apply_fun r2s x) (minus_SNo one_third)) Hr2s_bounds).
L176153
We prove the intermediate claim Hr3Rx: add_SNo (apply_fun r2s x) one_third ∈ R.
L176155
An exact proof term for the current goal is (real_add_SNo (apply_fun r2s x) Hr2sRx one_third H13R).
L176155
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r2s x) one_third).
L176157
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r2s x) one_third Hm1R Hr2sRx H13R Hm1le).
L176157
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r2s x) one_third).
L176159
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L176159
An exact proof term for the current goal is Hlow_tmp.
L176160
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r2s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L176162
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r2s x) (minus_SNo one_third) one_third Hr2sRx Hm13R H13R HhiI1).
L176162
We prove the intermediate claim H13S: SNo one_third.
L176164
An exact proof term for the current goal is (real_SNo one_third H13R).
L176164
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r2s x) one_third) 0.
L176166
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L176166
An exact proof term for the current goal is Hup0_tmp.
L176167
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L176169
An exact proof term for the current goal is Rle_0_two_thirds.
L176169
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r2s x) one_third) two_thirds.
L176171
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r2s x) one_third) 0 two_thirds Hup0 H0le23).
L176171
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r2s x) one_third) Hm23R H23R Hr3Rx Hlow Hup).
L176175
Assume HxnotB3: ¬ (x ∈ B3).
L176175
We prove the intermediate claim HC3_cases: x ∈ C3 ∨ ¬ (x ∈ C3).
L176177
An exact proof term for the current goal is (xm (x ∈ C3)).
L176177
Apply (HC3_cases (apply_fun r3 x ∈ I2)) to the current goal.
L176179
Assume HxC3: x ∈ C3.
L176179
We prove the intermediate claim Hu3eq: apply_fun u3 x = one_third.
L176181
An exact proof term for the current goal is (Hu3_on_C3 x HxC3).
L176181
We prove the intermediate claim Hr3eq: apply_fun r3 x = add_SNo (apply_fun r2s x) (minus_SNo one_third).
L176183
rewrite the current goal using (Hr3_apply x HxA) (from left to right).
L176183
rewrite the current goal using Hu3eq (from left to right) at position 1.
Use reflexivity.
L176185
rewrite the current goal using Hr3eq (from left to right).
L176186
We prove the intermediate claim Hr2sI3I: apply_fun r2s x ∈ I3 ∩ I.
L176188
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r2s x0 ∈ I3 ∩ I) x HxC3).
L176188
We prove the intermediate claim Hr2sI3: apply_fun r2s x ∈ I3.
L176190
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r2s x) Hr2sI3I).
L176190
We prove the intermediate claim H13R: one_third ∈ R.
L176192
An exact proof term for the current goal is one_third_in_R.
L176192
We prove the intermediate claim H23R: two_thirds ∈ R.
L176194
An exact proof term for the current goal is two_thirds_in_R.
L176194
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L176196
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L176196
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L176198
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L176198
We prove the intermediate claim Hr2sRx: apply_fun r2s x ∈ R.
L176200
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r2s x) Hr2sI3).
L176200
We prove the intermediate claim Hr2s_bounds: Rle one_third (apply_fun r2s x) ∧ Rle (apply_fun r2s x) 1.
L176202
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r2s x) H13R real_1 Hr2sI3).
L176202
We prove the intermediate claim HloI3: Rle one_third (apply_fun r2s x).
L176204
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r2s x)) (Rle (apply_fun r2s x) 1) Hr2s_bounds).
L176204
We prove the intermediate claim HhiI3: Rle (apply_fun r2s x) 1.
L176206
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r2s x)) (Rle (apply_fun r2s x) 1) Hr2s_bounds).
L176206
We prove the intermediate claim Hr3Rx: add_SNo (apply_fun r2s x) (minus_SNo one_third) ∈ R.
L176208
An exact proof term for the current goal is (real_add_SNo (apply_fun r2s x) Hr2sRx (minus_SNo one_third) Hm13R).
L176208
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r2s x) (minus_SNo one_third)).
L176211
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r2s x) (minus_SNo one_third) H13R Hr2sRx Hm13R HloI3).
L176212
We prove the intermediate claim H13S: SNo one_third.
L176214
An exact proof term for the current goal is (real_SNo one_third H13R).
L176214
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r2s x) (minus_SNo one_third)).
L176216
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L176216
An exact proof term for the current goal is H0le_tmp.
L176217
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L176219
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L176219
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r2s x) (minus_SNo one_third)).
L176221
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r2s x) (minus_SNo one_third)) Hm23le0 H0le).
L176223
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r2s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L176226
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r2s x) 1 (minus_SNo one_third) Hr2sRx real_1 Hm13R HhiI3).
L176227
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L176229
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L176229
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L176230
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r2s x) (minus_SNo one_third)) two_thirds.
L176232
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r2s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L176235
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r2s x) (minus_SNo one_third)) Hm23R H23R Hr3Rx Hlow Hup).
L176239
Assume HxnotC3: ¬ (x ∈ C3).
L176239
We prove the intermediate claim HxX: x ∈ X.
L176241
An exact proof term for the current goal is (HAsubX x HxA).
L176241
We prove the intermediate claim HnotI1: ¬ (apply_fun r2s x ∈ I1).
L176243
Assume Hr2sI1': apply_fun r2s x ∈ I1.
L176243
We prove the intermediate claim Hr2sI1I: apply_fun r2s x ∈ I1 ∩ I.
L176245
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r2s x) Hr2sI1' Hr2sIx).
L176245
We prove the intermediate claim HxB3': x ∈ B3.
L176247
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r2s x0 ∈ I1 ∩ I) x HxA Hr2sI1I).
L176247
Apply FalseE to the current goal.
L176248
An exact proof term for the current goal is (HxnotB3 HxB3').
L176249
We prove the intermediate claim HnotI3: ¬ (apply_fun r2s x ∈ I3).
L176251
Assume Hr2sI3': apply_fun r2s x ∈ I3.
L176251
We prove the intermediate claim Hr2sI3I: apply_fun r2s x ∈ I3 ∩ I.
L176253
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r2s x) Hr2sI3' Hr2sIx).
L176253
We prove the intermediate claim HxC3': x ∈ C3.
L176255
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r2s x0 ∈ I3 ∩ I) x HxA Hr2sI3I).
L176255
Apply FalseE to the current goal.
L176256
An exact proof term for the current goal is (HxnotC3 HxC3').
L176257
We prove the intermediate claim Hr2sRx: apply_fun r2s x ∈ R.
L176259
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r2s x) Hr2sIx).
L176259
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L176261
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L176261
We prove the intermediate claim H13R: one_third ∈ R.
L176263
An exact proof term for the current goal is one_third_in_R.
L176263
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L176265
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L176265
We prove the intermediate claim Hr2s_boundsI: Rle (minus_SNo 1) (apply_fun r2s x) ∧ Rle (apply_fun r2s x) 1.
L176267
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r2s x) Hm1R real_1 Hr2sIx).
L176267
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r2s x) (minus_SNo 1)).
L176269
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r2s x) (andEL (Rle (minus_SNo 1) (apply_fun r2s x)) (Rle (apply_fun r2s x) 1) Hr2s_boundsI)).
L176270
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r2s x)).
L176272
An exact proof term for the current goal is (RleE_nlt (apply_fun r2s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r2s x)) (Rle (apply_fun r2s x) 1) Hr2s_boundsI)).
L176273
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r2s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r2s x).
L176275
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r2s x) Hm1R Hm13R Hr2sRx HnotI1).
L176276
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r2s x).
L176278
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r2s x))) to the current goal.
L176279
Assume Hbad: Rlt (apply_fun r2s x) (minus_SNo 1).
L176279
Apply FalseE to the current goal.
L176280
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L176282
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r2s x).
L176282
An exact proof term for the current goal is Hok.
L176283
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r2s x) (minus_SNo one_third)).
L176285
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r2s x) Hm13lt_fx).
L176285
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r2s x) one_third ∨ Rlt 1 (apply_fun r2s x).
L176287
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r2s x) H13R real_1 Hr2sRx HnotI3).
L176288
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r2s x) one_third.
L176290
Apply (HnotI3_cases (Rlt (apply_fun r2s x) one_third)) to the current goal.
L176291
Assume Hok: Rlt (apply_fun r2s x) one_third.
L176291
An exact proof term for the current goal is Hok.
L176293
Assume Hbad: Rlt 1 (apply_fun r2s x).
L176293
Apply FalseE to the current goal.
L176294
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L176295
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r2s x)).
L176297
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r2s x) one_third Hfx_lt_13).
L176297
We prove the intermediate claim Hr2sI0: apply_fun r2s x ∈ I0.
L176299
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L176301
We prove the intermediate claim HxSep: apply_fun r2s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L176303
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r2s x) Hr2sRx (andI (¬ (Rlt (apply_fun r2s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r2s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L176307
rewrite the current goal using HI0_def (from left to right).
L176308
An exact proof term for the current goal is HxSep.
L176309
We prove the intermediate claim Hu3funI0: function_on u3 X I0.
L176311
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u3 Hu3contI0).
L176311
We prove the intermediate claim Hu3xI0: apply_fun u3 x ∈ I0.
L176313
An exact proof term for the current goal is (Hu3funI0 x HxX).
L176313
We prove the intermediate claim Hu3xR: apply_fun u3 x ∈ R.
L176315
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u3 x) Hu3xI0).
L176315
We prove the intermediate claim Hm_u3x_R: minus_SNo (apply_fun u3 x) ∈ R.
L176317
An exact proof term for the current goal is (real_minus_SNo (apply_fun u3 x) Hu3xR).
L176317
rewrite the current goal using (Hr3_apply x HxA) (from left to right).
L176318
We prove the intermediate claim Hr3xR: add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x)) ∈ R.
L176320
An exact proof term for the current goal is (real_add_SNo (apply_fun r2s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r2s x) Hr2sI0) (minus_SNo (apply_fun u3 x)) Hm_u3x_R).
L176322
We prove the intermediate claim H23R: two_thirds ∈ R.
L176324
An exact proof term for the current goal is two_thirds_in_R.
L176324
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L176326
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L176326
We prove the intermediate claim Hr2s_bounds0: Rle (minus_SNo one_third) (apply_fun r2s x) ∧ Rle (apply_fun r2s x) one_third.
L176328
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r2s x) Hm13R H13R Hr2sI0).
L176328
We prove the intermediate claim Hu3_bounds0: Rle (minus_SNo one_third) (apply_fun u3 x) ∧ Rle (apply_fun u3 x) one_third.
L176330
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u3 x) Hm13R H13R Hu3xI0).
L176330
We prove the intermediate claim Hm13_le_r2s: Rle (minus_SNo one_third) (apply_fun r2s x).
L176332
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r2s x)) (Rle (apply_fun r2s x) one_third) Hr2s_bounds0).
L176332
We prove the intermediate claim Hr2s_le_13: Rle (apply_fun r2s x) one_third.
L176334
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r2s x)) (Rle (apply_fun r2s x) one_third) Hr2s_bounds0).
L176334
We prove the intermediate claim Hm13_le_u3x: Rle (minus_SNo one_third) (apply_fun u3 x).
L176336
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u3 x)) (Rle (apply_fun u3 x) one_third) Hu3_bounds0).
L176336
We prove the intermediate claim Hu3x_le_13: Rle (apply_fun u3 x) one_third.
L176338
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u3 x)) (Rle (apply_fun u3 x) one_third) Hu3_bounds0).
L176338
We prove the intermediate claim Hm13_le_mu3: Rle (minus_SNo one_third) (minus_SNo (apply_fun u3 x)).
L176340
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u3 x) one_third Hu3x_le_13).
L176340
We prove the intermediate claim Hmu3_le_13: Rle (minus_SNo (apply_fun u3 x)) one_third.
L176342
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u3 x)) (minus_SNo (minus_SNo one_third)).
L176343
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u3 x) Hm13_le_u3x).
L176343
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L176344
An exact proof term for the current goal is Htmp.
L176345
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u3 x))).
L176348
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u3 x)) Hm13R Hm13R Hm_u3x_R Hm13_le_mu3).
L176349
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u3 x))) (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))).
L176352
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r2s x) (minus_SNo (apply_fun u3 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r2s x) Hr2sI0) Hm_u3x_R Hm13_le_r2s).
L176356
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))).
L176359
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u3 x))) (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))) Hlow1 Hlow2).
L176362
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))).
L176364
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L176364
An exact proof term for the current goal is Hlow_tmp.
L176365
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))) (add_SNo (apply_fun r2s x) one_third).
L176368
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r2s x) (minus_SNo (apply_fun u3 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r2s x) Hr2sI0) Hm_u3x_R H13R Hmu3_le_13).
L176370
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r2s x) one_third) (add_SNo one_third one_third).
L176373
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r2s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r2s x) Hr2sI0) H13R H13R Hr2s_le_13).
L176375
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))) (add_SNo one_third one_third).
L176378
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))) (add_SNo (apply_fun r2s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L176381
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L176383
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))) two_thirds.
L176385
rewrite the current goal using Hdef23 (from left to right) at position 1.
L176385
An exact proof term for the current goal is Hup_tmp.
L176386
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))) Hm23R H23R Hr3xR Hlow Hup).
L176389
Set r3s to be the term compose_fun A r3 (div_const_fun den).
L176390
We prove the intermediate claim Hr3s_cont: continuous_map A Ta I Ti r3s.
L176392
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
L176393
An exact proof term for the current goal is R_standard_topology_is_topology.
L176393
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L176395
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L176395
We prove the intermediate claim Hr3s_contR: continuous_map A Ta R R_standard_topology r3s.
L176397
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r3 (div_const_fun den) Hr3_cont Hdivcont).
L176398
We prove the intermediate claim Hr3s_I: ∀x : set, x ∈ A → apply_fun r3s x ∈ I.
L176400
Let x be given.
L176400
Assume HxA: x ∈ A.
L176400
We prove the intermediate claim Hr3xI2: apply_fun r3 x ∈ I2.
L176402
An exact proof term for the current goal is (Hr3_range x HxA).
L176402
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L176404
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L176404
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L176406
An exact proof term for the current goal is (real_minus_SNo den H23R).
L176406
We prove the intermediate claim Hr3xR: apply_fun r3 x ∈ R.
L176408
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r3 x) Hr3xI2).
L176408
We prove the intermediate claim Hr3xS: SNo (apply_fun r3 x).
L176410
An exact proof term for the current goal is (real_SNo (apply_fun r3 x) Hr3xR).
L176410
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r3 x) ∧ Rle (apply_fun r3 x) den.
L176412
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r3 x) HmdenR H23R Hr3xI2).
L176412
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r3 x).
L176414
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r3 x)) (Rle (apply_fun r3 x) den) Hbounds).
L176416
We prove the intermediate claim Hhi: Rle (apply_fun r3 x) den.
L176418
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r3 x)) (Rle (apply_fun r3 x) den) Hbounds).
L176420
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r3 x)).
L176422
An exact proof term for the current goal is (RleE_nlt (apply_fun r3 x) den Hhi).
L176422
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r3 x) (minus_SNo den)).
L176424
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r3 x) Hlo).
L176424
We prove the intermediate claim HyEq: apply_fun r3s x = div_SNo (apply_fun r3 x) den.
L176426
rewrite the current goal using (compose_fun_apply A r3 (div_const_fun den) x HxA) (from left to right).
L176426
rewrite the current goal using (div_const_fun_apply den (apply_fun r3 x) H23R Hr3xR) (from left to right).
Use reflexivity.
L176428
We prove the intermediate claim HyR: apply_fun r3s x ∈ R.
L176430
rewrite the current goal using HyEq (from left to right).
L176430
An exact proof term for the current goal is (real_div_SNo (apply_fun r3 x) Hr3xR den H23R).
L176431
We prove the intermediate claim HyS: SNo (apply_fun r3s x).
L176433
An exact proof term for the current goal is (real_SNo (apply_fun r3s x) HyR).
L176433
We prove the intermediate claim Hy_le_1: Rle (apply_fun r3s x) 1.
L176435
Apply (RleI (apply_fun r3s x) 1 HyR real_1) to the current goal.
L176435
We will prove ¬ (Rlt 1 (apply_fun r3s x)).
L176436
Assume H1lt: Rlt 1 (apply_fun r3s x).
L176437
We prove the intermediate claim H1lty: 1 < apply_fun r3s x.
L176439
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r3s x) H1lt).
L176439
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r3s x) den.
L176441
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r3s x) den SNo_1 HyS H23S H23pos H1lty).
L176441
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r3s x) den.
L176443
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L176443
An exact proof term for the current goal is HmulLt.
L176444
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r3s x) den = apply_fun r3 x.
L176446
rewrite the current goal using HyEq (from left to right) at position 1.
L176446
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r3 x) den Hr3xS H23S H23ne0).
L176447
We prove the intermediate claim Hden_lt_r3x: den < apply_fun r3 x.
L176449
rewrite the current goal using HmulEq (from right to left).
L176449
An exact proof term for the current goal is HmulLt'.
L176450
We prove the intermediate claim Hbad: Rlt den (apply_fun r3 x).
L176452
An exact proof term for the current goal is (RltI den (apply_fun r3 x) H23R Hr3xR Hden_lt_r3x).
L176452
An exact proof term for the current goal is (Hnlt_hi Hbad).
L176453
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r3s x).
L176455
Apply (RleI (minus_SNo 1) (apply_fun r3s x) Hm1R HyR) to the current goal.
L176455
We will prove ¬ (Rlt (apply_fun r3s x) (minus_SNo 1)).
L176456
Assume Hylt: Rlt (apply_fun r3s x) (minus_SNo 1).
L176457
We prove the intermediate claim Hylts: apply_fun r3s x < minus_SNo 1.
L176459
An exact proof term for the current goal is (RltE_lt (apply_fun r3s x) (minus_SNo 1) Hylt).
L176459
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r3s x) den < mul_SNo (minus_SNo 1) den.
L176461
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r3s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L176462
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r3s x) den = apply_fun r3 x.
L176464
rewrite the current goal using HyEq (from left to right) at position 1.
L176464
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r3 x) den Hr3xS H23S H23ne0).
L176465
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L176467
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L176467
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L176469
We prove the intermediate claim Hr3x_lt_mden: apply_fun r3 x < minus_SNo den.
L176471
rewrite the current goal using HmulEq (from right to left).
L176471
rewrite the current goal using HrhsEq (from right to left).
L176472
An exact proof term for the current goal is HmulLt.
L176473
We prove the intermediate claim Hbad: Rlt (apply_fun r3 x) (minus_SNo den).
L176475
An exact proof term for the current goal is (RltI (apply_fun r3 x) (minus_SNo den) Hr3xR HmdenR Hr3x_lt_mden).
L176475
An exact proof term for the current goal is (Hnlt_lo Hbad).
L176476
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r3s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L176478
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r3s I Hr3s_contR HIcR Hr3s_I).
L176479
We prove the intermediate claim Hex_u4: ∃u4 : set, continuous_map X Tx I0 T0 u4 ∧ (∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r3s ((closed_interval one_third 1) ∩ I) → apply_fun u4 x = one_third).
(*** fifth correction step scaffold: build u4 and g4 from residual r3s ***)
L176489
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r3s Hnorm HA Hr3s_cont).
L176489
Apply Hex_u4 to the current goal.
L176490
Let u4 be given.
L176491
Assume Hu4.
L176491
We prove the intermediate claim Hu4contI0: continuous_map X Tx I0 T0 u4.
L176493
We prove the intermediate claim Hu4AB: continuous_map X Tx I0 T0 u4 ∧ (∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third).
L176497
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u4 ∧ (∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r3s ((closed_interval one_third 1) ∩ I) → apply_fun u4 x = one_third) Hu4).
L176503
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u4) (∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third) Hu4AB).
L176508
We prove the intermediate claim Hu4contR: continuous_map X Tx R R_standard_topology u4.
L176510
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L176511
We prove the intermediate claim HI0subR: I0 ⊆ R.
L176513
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L176513
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u4 Hu4contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L176521
Set den4 to be the term mul_SNo den3 den.
L176522
We prove the intermediate claim Hden4R: den4 ∈ R.
L176524
An exact proof term for the current goal is (real_mul_SNo den3 Hden3R den H23R).
L176524
We prove the intermediate claim Hden4pos: 0 < den4.
L176526
We prove the intermediate claim Hden3S: SNo den3.
L176527
An exact proof term for the current goal is (real_SNo den3 Hden3R).
L176527
An exact proof term for the current goal is (mul_SNo_pos_pos den3 den Hden3S H23S Hden3pos HdenPos).
L176528
Set u4s to be the term compose_fun X u4 (mul_const_fun den4).
L176529
We prove the intermediate claim Hu4s_cont: continuous_map X Tx R R_standard_topology u4s.
L176531
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u4 den4 HTx Hu4contR Hden4R Hden4pos).
L176531
Set g4 to be the term compose_fun X (pair_map X g3 u4s) add_fun_R.
L176532
We prove the intermediate claim Hg4cont: continuous_map X Tx R R_standard_topology g4.
L176534
An exact proof term for the current goal is (add_two_continuous_R X Tx g3 u4s HTx Hg3cont Hu4s_cont).
L176534
We prove the intermediate claim Hu4contA: continuous_map A Ta R R_standard_topology u4.
(*** sixth correction step scaffold: residual r4 on A and scaling r4s ***)
L176538
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u4 A HTx HAsubX Hu4contR).
L176538
Set u4neg to be the term compose_fun A u4 neg_fun.
L176539
We prove the intermediate claim Hu4neg_cont: continuous_map A Ta R R_standard_topology u4neg.
L176541
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u4 neg_fun Hu4contA Hnegcont).
L176542
We prove the intermediate claim Hr3s_contR: continuous_map A Ta R R_standard_topology r3s.
L176544
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r3s Hr3s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L176545
Set r4 to be the term compose_fun A (pair_map A r3s u4neg) add_fun_R.
L176546
We prove the intermediate claim Hr4_cont: continuous_map A Ta R R_standard_topology r4.
L176548
An exact proof term for the current goal is (add_two_continuous_R A Ta r3s u4neg HTa Hr3s_contR Hu4neg_cont).
L176548
We prove the intermediate claim Hr4_apply: ∀x : set, x ∈ A → apply_fun r4 x = add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x)).
L176551
Let x be given.
L176551
Assume HxA: x ∈ A.
L176551
We prove the intermediate claim Hpimg: apply_fun (pair_map A r3s u4neg) x ∈ setprod R R.
L176553
rewrite the current goal using (pair_map_apply A R R r3s u4neg x HxA) (from left to right).
L176553
We prove the intermediate claim Hr3sxI: apply_fun r3s x ∈ I.
L176555
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r3s Hr3s_cont x HxA).
L176555
We prove the intermediate claim Hr3sxR: apply_fun r3s x ∈ R.
L176557
An exact proof term for the current goal is (HIcR (apply_fun r3s x) Hr3sxI).
L176557
We prove the intermediate claim Hu4negRx: apply_fun u4neg x ∈ R.
L176559
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u4neg Hu4neg_cont x HxA).
L176559
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r3s x) (apply_fun u4neg x) Hr3sxR Hu4negRx).
L176560
rewrite the current goal using (compose_fun_apply A (pair_map A r3s u4neg) add_fun_R x HxA) (from left to right).
L176561
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r3s u4neg) x) Hpimg) (from left to right) at position 1.
L176562
rewrite the current goal using (pair_map_apply A R R r3s u4neg x HxA) (from left to right).
L176563
rewrite the current goal using (tuple_2_0_eq (apply_fun r3s x) (apply_fun u4neg x)) (from left to right).
L176564
rewrite the current goal using (tuple_2_1_eq (apply_fun r3s x) (apply_fun u4neg x)) (from left to right).
L176565
rewrite the current goal using (compose_fun_apply A u4 neg_fun x HxA) (from left to right) at position 1.
L176566
We prove the intermediate claim Hu4Rx: apply_fun u4 x ∈ R.
L176568
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u4 Hu4contA x HxA).
L176568
rewrite the current goal using (neg_fun_apply (apply_fun u4 x) Hu4Rx) (from left to right) at position 1.
Use reflexivity.
L176570
We prove the intermediate claim Hr4_range: ∀x : set, x ∈ A → apply_fun r4 x ∈ I2.
L176572
We prove the intermediate claim Hu4AB: continuous_map X Tx I0 T0 u4 ∧ (∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third).
(*** show the next residual stays within [-2/3,2/3] on A ***)
L176577
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u4 ∧ (∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r3s ((closed_interval one_third 1) ∩ I) → apply_fun u4 x = one_third) Hu4).
L176583
We prove the intermediate claim Hu4_on_B4: ∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third.
L176587
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u4) (∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third) Hu4AB).
L176591
We prove the intermediate claim Hu4_on_C4: ∀x : set, x ∈ preimage_of A r3s ((closed_interval one_third 1) ∩ I) → apply_fun u4 x = one_third.
L176595
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u4 ∧ (∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r3s ((closed_interval one_third 1) ∩ I) → apply_fun u4 x = one_third) Hu4).
L176601
Let x be given.
L176602
Assume HxA: x ∈ A.
L176602
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L176603
Set I3 to be the term closed_interval one_third 1.
L176604
Set B4 to be the term preimage_of A r3s (I1 ∩ I).
L176605
Set C4 to be the term preimage_of A r3s (I3 ∩ I).
L176606
We prove the intermediate claim Hr3sIx: apply_fun r3s x ∈ I.
L176608
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r3s Hr3s_cont x HxA).
L176608
We prove the intermediate claim HB4_cases: x ∈ B4 ∨ ¬ (x ∈ B4).
L176610
An exact proof term for the current goal is (xm (x ∈ B4)).
L176610
Apply (HB4_cases (apply_fun r4 x ∈ I2)) to the current goal.
L176612
Assume HxB4: x ∈ B4.
L176612
We prove the intermediate claim Hu4eq: apply_fun u4 x = minus_SNo one_third.
L176614
An exact proof term for the current goal is (Hu4_on_B4 x HxB4).
L176614
We prove the intermediate claim Hr4eq: apply_fun r4 x = add_SNo (apply_fun r3s x) one_third.
L176616
rewrite the current goal using (Hr4_apply x HxA) (from left to right).
L176616
rewrite the current goal using Hu4eq (from left to right) at position 1.
L176617
We prove the intermediate claim H13R: one_third ∈ R.
L176619
An exact proof term for the current goal is one_third_in_R.
L176619
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L176621
rewrite the current goal using Hr4eq (from left to right).
L176622
We prove the intermediate claim Hr3sI1I: apply_fun r3s x ∈ I1 ∩ I.
L176624
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r3s x0 ∈ I1 ∩ I) x HxB4).
L176624
We prove the intermediate claim Hr3sI1: apply_fun r3s x ∈ I1.
L176626
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r3s x) Hr3sI1I).
L176626
We prove the intermediate claim H13R: one_third ∈ R.
L176628
An exact proof term for the current goal is one_third_in_R.
L176628
We prove the intermediate claim H23R: two_thirds ∈ R.
L176630
An exact proof term for the current goal is two_thirds_in_R.
L176630
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L176632
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L176632
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L176634
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L176634
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L176636
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L176636
We prove the intermediate claim Hr3sRx: apply_fun r3s x ∈ R.
L176638
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r3s x) Hr3sI1).
L176638
We prove the intermediate claim Hr3s_bounds: Rle (minus_SNo 1) (apply_fun r3s x) ∧ Rle (apply_fun r3s x) (minus_SNo one_third).
L176640
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r3s x) Hm1R Hm13R Hr3sI1).
L176641
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r3s x).
L176643
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r3s x)) (Rle (apply_fun r3s x) (minus_SNo one_third)) Hr3s_bounds).
L176644
We prove the intermediate claim HhiI1: Rle (apply_fun r3s x) (minus_SNo one_third).
L176646
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r3s x)) (Rle (apply_fun r3s x) (minus_SNo one_third)) Hr3s_bounds).
L176647
We prove the intermediate claim Hr4Rx: add_SNo (apply_fun r3s x) one_third ∈ R.
L176649
An exact proof term for the current goal is (real_add_SNo (apply_fun r3s x) Hr3sRx one_third H13R).
L176649
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r3s x) one_third).
L176651
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r3s x) one_third Hm1R Hr3sRx H13R Hm1le).
L176651
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r3s x) one_third).
L176653
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L176653
An exact proof term for the current goal is Hlow_tmp.
L176654
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r3s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L176656
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r3s x) (minus_SNo one_third) one_third Hr3sRx Hm13R H13R HhiI1).
L176656
We prove the intermediate claim H13S: SNo one_third.
L176658
An exact proof term for the current goal is (real_SNo one_third H13R).
L176658
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r3s x) one_third) 0.
L176660
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L176660
An exact proof term for the current goal is Hup0_tmp.
L176661
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L176663
An exact proof term for the current goal is Rle_0_two_thirds.
L176663
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r3s x) one_third) two_thirds.
L176665
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r3s x) one_third) 0 two_thirds Hup0 H0le23).
L176665
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r3s x) one_third) Hm23R H23R Hr4Rx Hlow Hup).
L176669
Assume HxnotB4: ¬ (x ∈ B4).
L176669
We prove the intermediate claim HC4_cases: x ∈ C4 ∨ ¬ (x ∈ C4).
L176671
An exact proof term for the current goal is (xm (x ∈ C4)).
L176671
Apply (HC4_cases (apply_fun r4 x ∈ I2)) to the current goal.
L176673
Assume HxC4: x ∈ C4.
L176673
We prove the intermediate claim Hu4eq: apply_fun u4 x = one_third.
L176675
An exact proof term for the current goal is (Hu4_on_C4 x HxC4).
L176675
We prove the intermediate claim Hr4eq: apply_fun r4 x = add_SNo (apply_fun r3s x) (minus_SNo one_third).
L176677
rewrite the current goal using (Hr4_apply x HxA) (from left to right).
L176677
rewrite the current goal using Hu4eq (from left to right) at position 1.
Use reflexivity.
L176679
rewrite the current goal using Hr4eq (from left to right).
L176680
We prove the intermediate claim Hr3sI3I: apply_fun r3s x ∈ I3 ∩ I.
L176682
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r3s x0 ∈ I3 ∩ I) x HxC4).
L176682
We prove the intermediate claim Hr3sI3: apply_fun r3s x ∈ I3.
L176684
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r3s x) Hr3sI3I).
L176684
We prove the intermediate claim H13R: one_third ∈ R.
L176686
An exact proof term for the current goal is one_third_in_R.
L176686
We prove the intermediate claim H23R: two_thirds ∈ R.
L176688
An exact proof term for the current goal is two_thirds_in_R.
L176688
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L176690
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L176690
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L176692
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L176692
We prove the intermediate claim Hr3sRx: apply_fun r3s x ∈ R.
L176694
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r3s x) Hr3sI3).
L176694
We prove the intermediate claim Hr3s_bounds: Rle one_third (apply_fun r3s x) ∧ Rle (apply_fun r3s x) 1.
L176696
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r3s x) H13R real_1 Hr3sI3).
L176696
We prove the intermediate claim HloI3: Rle one_third (apply_fun r3s x).
L176698
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r3s x)) (Rle (apply_fun r3s x) 1) Hr3s_bounds).
L176698
We prove the intermediate claim HhiI3: Rle (apply_fun r3s x) 1.
L176700
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r3s x)) (Rle (apply_fun r3s x) 1) Hr3s_bounds).
L176700
We prove the intermediate claim Hr4Rx: add_SNo (apply_fun r3s x) (minus_SNo one_third) ∈ R.
L176702
An exact proof term for the current goal is (real_add_SNo (apply_fun r3s x) Hr3sRx (minus_SNo one_third) Hm13R).
L176702
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r3s x) (minus_SNo one_third)).
L176705
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r3s x) (minus_SNo one_third) H13R Hr3sRx Hm13R HloI3).
L176706
We prove the intermediate claim H13S: SNo one_third.
L176708
An exact proof term for the current goal is (real_SNo one_third H13R).
L176708
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r3s x) (minus_SNo one_third)).
L176710
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L176710
An exact proof term for the current goal is H0le_tmp.
L176711
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L176713
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L176713
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r3s x) (minus_SNo one_third)).
L176715
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r3s x) (minus_SNo one_third)) Hm23le0 H0le).
L176717
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r3s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L176720
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r3s x) 1 (minus_SNo one_third) Hr3sRx real_1 Hm13R HhiI3).
L176721
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L176723
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L176723
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L176724
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r3s x) (minus_SNo one_third)) two_thirds.
L176726
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r3s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L176729
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r3s x) (minus_SNo one_third)) Hm23R H23R Hr4Rx Hlow Hup).
L176733
Assume HxnotC4: ¬ (x ∈ C4).
L176733
We prove the intermediate claim HxX: x ∈ X.
L176735
An exact proof term for the current goal is (HAsubX x HxA).
L176735
We prove the intermediate claim HnotI1: ¬ (apply_fun r3s x ∈ I1).
L176737
Assume Hr3sI1': apply_fun r3s x ∈ I1.
L176737
We prove the intermediate claim Hr3sI1I: apply_fun r3s x ∈ I1 ∩ I.
L176739
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r3s x) Hr3sI1' Hr3sIx).
L176739
We prove the intermediate claim HxB4': x ∈ B4.
L176741
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r3s x0 ∈ I1 ∩ I) x HxA Hr3sI1I).
L176741
Apply FalseE to the current goal.
L176742
An exact proof term for the current goal is (HxnotB4 HxB4').
L176743
We prove the intermediate claim HnotI3: ¬ (apply_fun r3s x ∈ I3).
L176745
Assume Hr3sI3': apply_fun r3s x ∈ I3.
L176745
We prove the intermediate claim Hr3sI3I: apply_fun r3s x ∈ I3 ∩ I.
L176747
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r3s x) Hr3sI3' Hr3sIx).
L176747
We prove the intermediate claim HxC4': x ∈ C4.
L176749
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r3s x0 ∈ I3 ∩ I) x HxA Hr3sI3I).
L176749
Apply FalseE to the current goal.
L176750
An exact proof term for the current goal is (HxnotC4 HxC4').
L176751
We prove the intermediate claim Hr3sRx: apply_fun r3s x ∈ R.
L176753
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r3s x) Hr3sIx).
L176753
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L176755
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L176755
We prove the intermediate claim H13R: one_third ∈ R.
L176757
An exact proof term for the current goal is one_third_in_R.
L176757
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L176759
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L176759
We prove the intermediate claim Hr3s_boundsI: Rle (minus_SNo 1) (apply_fun r3s x) ∧ Rle (apply_fun r3s x) 1.
L176761
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r3s x) Hm1R real_1 Hr3sIx).
L176761
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r3s x) (minus_SNo 1)).
L176763
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r3s x) (andEL (Rle (minus_SNo 1) (apply_fun r3s x)) (Rle (apply_fun r3s x) 1) Hr3s_boundsI)).
L176764
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r3s x)).
L176766
An exact proof term for the current goal is (RleE_nlt (apply_fun r3s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r3s x)) (Rle (apply_fun r3s x) 1) Hr3s_boundsI)).
L176767
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r3s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r3s x).
L176769
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r3s x) Hm1R Hm13R Hr3sRx HnotI1).
L176770
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r3s x).
L176772
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r3s x))) to the current goal.
L176773
Assume Hbad: Rlt (apply_fun r3s x) (minus_SNo 1).
L176773
Apply FalseE to the current goal.
L176774
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L176776
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r3s x).
L176776
An exact proof term for the current goal is Hok.
L176777
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r3s x) (minus_SNo one_third)).
L176779
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r3s x) Hm13lt_fx).
L176779
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r3s x) one_third ∨ Rlt 1 (apply_fun r3s x).
L176781
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r3s x) H13R real_1 Hr3sRx HnotI3).
L176782
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r3s x) one_third.
L176784
Apply (HnotI3_cases (Rlt (apply_fun r3s x) one_third)) to the current goal.
L176785
Assume Hok: Rlt (apply_fun r3s x) one_third.
L176785
An exact proof term for the current goal is Hok.
L176787
Assume Hbad: Rlt 1 (apply_fun r3s x).
L176787
Apply FalseE to the current goal.
L176788
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L176789
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r3s x)).
L176791
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r3s x) one_third Hfx_lt_13).
L176791
We prove the intermediate claim Hr3sI0: apply_fun r3s x ∈ I0.
L176793
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L176795
We prove the intermediate claim HxSep: apply_fun r3s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L176797
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r3s x) Hr3sRx (andI (¬ (Rlt (apply_fun r3s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r3s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L176801
rewrite the current goal using HI0_def (from left to right).
L176802
An exact proof term for the current goal is HxSep.
L176803
We prove the intermediate claim Hu4funI0: function_on u4 X I0.
L176805
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u4 Hu4contI0).
L176805
We prove the intermediate claim Hu4xI0: apply_fun u4 x ∈ I0.
L176807
An exact proof term for the current goal is (Hu4funI0 x HxX).
L176807
We prove the intermediate claim Hu4xR: apply_fun u4 x ∈ R.
L176809
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u4 x) Hu4xI0).
L176809
We prove the intermediate claim Hm_u4x_R: minus_SNo (apply_fun u4 x) ∈ R.
L176811
An exact proof term for the current goal is (real_minus_SNo (apply_fun u4 x) Hu4xR).
L176811
rewrite the current goal using (Hr4_apply x HxA) (from left to right).
L176812
We prove the intermediate claim Hr4xR: add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x)) ∈ R.
L176814
An exact proof term for the current goal is (real_add_SNo (apply_fun r3s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r3s x) Hr3sI0) (minus_SNo (apply_fun u4 x)) Hm_u4x_R).
L176816
We prove the intermediate claim H23R: two_thirds ∈ R.
L176818
An exact proof term for the current goal is two_thirds_in_R.
L176818
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L176820
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L176820
We prove the intermediate claim Hr3s_bounds0: Rle (minus_SNo one_third) (apply_fun r3s x) ∧ Rle (apply_fun r3s x) one_third.
L176822
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r3s x) Hm13R H13R Hr3sI0).
L176822
We prove the intermediate claim Hu4_bounds0: Rle (minus_SNo one_third) (apply_fun u4 x) ∧ Rle (apply_fun u4 x) one_third.
L176824
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u4 x) Hm13R H13R Hu4xI0).
L176824
We prove the intermediate claim Hm13_le_r3s: Rle (minus_SNo one_third) (apply_fun r3s x).
L176826
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r3s x)) (Rle (apply_fun r3s x) one_third) Hr3s_bounds0).
L176826
We prove the intermediate claim Hr3s_le_13: Rle (apply_fun r3s x) one_third.
L176828
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r3s x)) (Rle (apply_fun r3s x) one_third) Hr3s_bounds0).
L176828
We prove the intermediate claim Hm13_le_u4x: Rle (minus_SNo one_third) (apply_fun u4 x).
L176830
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u4 x)) (Rle (apply_fun u4 x) one_third) Hu4_bounds0).
L176830
We prove the intermediate claim Hu4x_le_13: Rle (apply_fun u4 x) one_third.
L176832
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u4 x)) (Rle (apply_fun u4 x) one_third) Hu4_bounds0).
L176832
We prove the intermediate claim Hm13_le_mu4: Rle (minus_SNo one_third) (minus_SNo (apply_fun u4 x)).
L176834
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u4 x) one_third Hu4x_le_13).
L176834
We prove the intermediate claim Hmu4_le_13: Rle (minus_SNo (apply_fun u4 x)) one_third.
L176836
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u4 x)) (minus_SNo (minus_SNo one_third)).
L176837
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u4 x) Hm13_le_u4x).
L176837
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L176838
An exact proof term for the current goal is Htmp.
L176839
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u4 x))).
L176842
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u4 x)) Hm13R Hm13R Hm_u4x_R Hm13_le_mu4).
L176843
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u4 x))) (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))).
L176846
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r3s x) (minus_SNo (apply_fun u4 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r3s x) Hr3sI0) Hm_u4x_R Hm13_le_r3s).
L176850
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))).
L176853
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u4 x))) (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))) Hlow1 Hlow2).
L176856
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))).
L176858
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L176858
An exact proof term for the current goal is Hlow_tmp.
L176859
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))) (add_SNo (apply_fun r3s x) one_third).
L176862
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r3s x) (minus_SNo (apply_fun u4 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r3s x) Hr3sI0) Hm_u4x_R H13R Hmu4_le_13).
L176864
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r3s x) one_third) (add_SNo one_third one_third).
L176867
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r3s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r3s x) Hr3sI0) H13R H13R Hr3s_le_13).
L176869
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))) (add_SNo one_third one_third).
L176872
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))) (add_SNo (apply_fun r3s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L176875
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L176877
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))) two_thirds.
L176879
rewrite the current goal using Hdef23 (from left to right) at position 1.
L176879
An exact proof term for the current goal is Hup_tmp.
L176880
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))) Hm23R H23R Hr4xR Hlow Hup).
L176883
Set r4s to be the term compose_fun A r4 (div_const_fun den).
L176884
We prove the intermediate claim Hr4s_cont: continuous_map A Ta I Ti r4s.
L176886
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
L176887
An exact proof term for the current goal is R_standard_topology_is_topology.
L176887
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L176889
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L176889
We prove the intermediate claim Hr4s_contR: continuous_map A Ta R R_standard_topology r4s.
L176891
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r4 (div_const_fun den) Hr4_cont Hdivcont).
L176892
We prove the intermediate claim Hr4s_I: ∀x : set, x ∈ A → apply_fun r4s x ∈ I.
L176894
Let x be given.
L176894
Assume HxA: x ∈ A.
L176894
We prove the intermediate claim Hr4xI2: apply_fun r4 x ∈ I2.
L176896
An exact proof term for the current goal is (Hr4_range x HxA).
L176896
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L176898
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L176898
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L176900
An exact proof term for the current goal is (real_minus_SNo den H23R).
L176900
We prove the intermediate claim Hr4xR: apply_fun r4 x ∈ R.
L176902
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r4 x) Hr4xI2).
L176902
We prove the intermediate claim Hr4xS: SNo (apply_fun r4 x).
L176904
An exact proof term for the current goal is (real_SNo (apply_fun r4 x) Hr4xR).
L176904
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r4 x) ∧ Rle (apply_fun r4 x) den.
L176906
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r4 x) HmdenR H23R Hr4xI2).
L176906
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r4 x).
L176908
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r4 x)) (Rle (apply_fun r4 x) den) Hbounds).
L176910
We prove the intermediate claim Hhi: Rle (apply_fun r4 x) den.
L176912
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r4 x)) (Rle (apply_fun r4 x) den) Hbounds).
L176914
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r4 x)).
L176916
An exact proof term for the current goal is (RleE_nlt (apply_fun r4 x) den Hhi).
L176916
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r4 x) (minus_SNo den)).
L176918
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r4 x) Hlo).
L176918
We prove the intermediate claim HyEq: apply_fun r4s x = div_SNo (apply_fun r4 x) den.
L176920
rewrite the current goal using (compose_fun_apply A r4 (div_const_fun den) x HxA) (from left to right).
L176920
rewrite the current goal using (div_const_fun_apply den (apply_fun r4 x) H23R Hr4xR) (from left to right).
Use reflexivity.
L176922
We prove the intermediate claim HyR: apply_fun r4s x ∈ R.
L176924
rewrite the current goal using HyEq (from left to right).
L176924
An exact proof term for the current goal is (real_div_SNo (apply_fun r4 x) Hr4xR den H23R).
L176925
We prove the intermediate claim HyS: SNo (apply_fun r4s x).
L176927
An exact proof term for the current goal is (real_SNo (apply_fun r4s x) HyR).
L176927
We prove the intermediate claim Hy_le_1: Rle (apply_fun r4s x) 1.
L176929
Apply (RleI (apply_fun r4s x) 1 HyR real_1) to the current goal.
L176929
We will prove ¬ (Rlt 1 (apply_fun r4s x)).
L176930
Assume H1lt: Rlt 1 (apply_fun r4s x).
L176931
We prove the intermediate claim H1lty: 1 < apply_fun r4s x.
L176933
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r4s x) H1lt).
L176933
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r4s x) den.
L176935
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r4s x) den SNo_1 HyS H23S H23pos H1lty).
L176935
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r4s x) den.
L176937
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L176937
An exact proof term for the current goal is HmulLt.
L176938
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r4s x) den = apply_fun r4 x.
L176940
rewrite the current goal using HyEq (from left to right) at position 1.
L176940
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r4 x) den Hr4xS H23S H23ne0).
L176941
We prove the intermediate claim Hden_lt_r4x: den < apply_fun r4 x.
L176943
rewrite the current goal using HmulEq (from right to left).
L176943
An exact proof term for the current goal is HmulLt'.
L176944
We prove the intermediate claim Hbad: Rlt den (apply_fun r4 x).
L176946
An exact proof term for the current goal is (RltI den (apply_fun r4 x) H23R Hr4xR Hden_lt_r4x).
L176946
An exact proof term for the current goal is (Hnlt_hi Hbad).
L176947
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r4s x).
L176949
Apply (RleI (minus_SNo 1) (apply_fun r4s x) Hm1R HyR) to the current goal.
L176949
We will prove ¬ (Rlt (apply_fun r4s x) (minus_SNo 1)).
L176950
Assume Hylt: Rlt (apply_fun r4s x) (minus_SNo 1).
L176951
We prove the intermediate claim Hylts: apply_fun r4s x < minus_SNo 1.
L176953
An exact proof term for the current goal is (RltE_lt (apply_fun r4s x) (minus_SNo 1) Hylt).
L176953
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r4s x) den < mul_SNo (minus_SNo 1) den.
L176955
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r4s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L176956
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r4s x) den = apply_fun r4 x.
L176958
rewrite the current goal using HyEq (from left to right) at position 1.
L176958
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r4 x) den Hr4xS H23S H23ne0).
L176959
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L176961
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L176961
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L176963
We prove the intermediate claim Hr4x_lt_mden: apply_fun r4 x < minus_SNo den.
L176965
rewrite the current goal using HmulEq (from right to left).
L176965
rewrite the current goal using HrhsEq (from right to left).
L176966
An exact proof term for the current goal is HmulLt.
L176967
We prove the intermediate claim Hbad: Rlt (apply_fun r4 x) (minus_SNo den).
L176969
An exact proof term for the current goal is (RltI (apply_fun r4 x) (minus_SNo den) Hr4xR HmdenR Hr4x_lt_mden).
L176969
An exact proof term for the current goal is (Hnlt_lo Hbad).
L176970
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r4s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L176972
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r4s I Hr4s_contR HIcR Hr4s_I).
L176973
We prove the intermediate claim Hex_u5: ∃u5 : set, continuous_map X Tx I0 T0 u5 ∧ (∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r4s ((closed_interval one_third 1) ∩ I) → apply_fun u5 x = one_third).
(*** seventh correction step scaffold: build u5 and g5 from residual r4s ***)
L176983
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r4s Hnorm HA Hr4s_cont).
L176983
Apply Hex_u5 to the current goal.
L176984
Let u5 be given.
L176985
Assume Hu5.
L176985
We prove the intermediate claim Hu5contI0: continuous_map X Tx I0 T0 u5.
L176987
We prove the intermediate claim Hu5AB: continuous_map X Tx I0 T0 u5 ∧ (∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third).
L176991
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u5 ∧ (∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r4s ((closed_interval one_third 1) ∩ I) → apply_fun u5 x = one_third) Hu5).
L176997
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u5) (∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third) Hu5AB).
L177002
We prove the intermediate claim Hu5contR: continuous_map X Tx R R_standard_topology u5.
L177004
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L177005
We prove the intermediate claim HI0subR: I0 ⊆ R.
L177007
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L177007
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u5 Hu5contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L177015
Set den5 to be the term mul_SNo den4 den.
L177016
We prove the intermediate claim Hden5R: den5 ∈ R.
L177018
An exact proof term for the current goal is (real_mul_SNo den4 Hden4R den H23R).
L177018
We prove the intermediate claim Hden5pos: 0 < den5.
L177020
We prove the intermediate claim Hden4S: SNo den4.
L177021
An exact proof term for the current goal is (real_SNo den4 Hden4R).
L177021
An exact proof term for the current goal is (mul_SNo_pos_pos den4 den Hden4S H23S Hden4pos HdenPos).
L177022
Set u5s to be the term compose_fun X u5 (mul_const_fun den5).
L177023
We prove the intermediate claim Hu5s_cont: continuous_map X Tx R R_standard_topology u5s.
L177025
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u5 den5 HTx Hu5contR Hden5R Hden5pos).
L177025
Set g5 to be the term compose_fun X (pair_map X g4 u5s) add_fun_R.
L177026
We prove the intermediate claim Hg5cont: continuous_map X Tx R R_standard_topology g5.
L177028
An exact proof term for the current goal is (add_two_continuous_R X Tx g4 u5s HTx Hg4cont Hu5s_cont).
L177028
We prove the intermediate claim Hu5contA: continuous_map A Ta R R_standard_topology u5.
(*** eighth correction step scaffold: residual r5 on A and scaling r5s ***)
L177032
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u5 A HTx HAsubX Hu5contR).
L177032
Set u5neg to be the term compose_fun A u5 neg_fun.
L177033
We prove the intermediate claim Hu5neg_cont: continuous_map A Ta R R_standard_topology u5neg.
L177035
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u5 neg_fun Hu5contA Hnegcont).
L177036
We prove the intermediate claim Hr4s_contR: continuous_map A Ta R R_standard_topology r4s.
L177038
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r4s Hr4s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L177039
Set r5 to be the term compose_fun A (pair_map A r4s u5neg) add_fun_R.
L177040
We prove the intermediate claim Hr5_cont: continuous_map A Ta R R_standard_topology r5.
L177042
An exact proof term for the current goal is (add_two_continuous_R A Ta r4s u5neg HTa Hr4s_contR Hu5neg_cont).
L177042
We prove the intermediate claim Hr5_apply: ∀x : set, x ∈ A → apply_fun r5 x = add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x)).
L177045
Let x be given.
L177045
Assume HxA: x ∈ A.
L177045
We prove the intermediate claim Hpimg: apply_fun (pair_map A r4s u5neg) x ∈ setprod R R.
L177047
rewrite the current goal using (pair_map_apply A R R r4s u5neg x HxA) (from left to right).
L177047
We prove the intermediate claim Hr4sxI: apply_fun r4s x ∈ I.
L177049
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r4s Hr4s_cont x HxA).
L177049
We prove the intermediate claim Hr4sxR: apply_fun r4s x ∈ R.
L177051
An exact proof term for the current goal is (HIcR (apply_fun r4s x) Hr4sxI).
L177051
We prove the intermediate claim Hu5negRx: apply_fun u5neg x ∈ R.
L177053
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u5neg Hu5neg_cont x HxA).
L177053
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r4s x) (apply_fun u5neg x) Hr4sxR Hu5negRx).
L177054
rewrite the current goal using (compose_fun_apply A (pair_map A r4s u5neg) add_fun_R x HxA) (from left to right).
L177055
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r4s u5neg) x) Hpimg) (from left to right) at position 1.
L177056
rewrite the current goal using (pair_map_apply A R R r4s u5neg x HxA) (from left to right).
L177057
rewrite the current goal using (tuple_2_0_eq (apply_fun r4s x) (apply_fun u5neg x)) (from left to right).
L177058
rewrite the current goal using (tuple_2_1_eq (apply_fun r4s x) (apply_fun u5neg x)) (from left to right).
L177059
rewrite the current goal using (compose_fun_apply A u5 neg_fun x HxA) (from left to right) at position 1.
L177060
We prove the intermediate claim Hu5Rx: apply_fun u5 x ∈ R.
L177062
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u5 Hu5contA x HxA).
L177062
rewrite the current goal using (neg_fun_apply (apply_fun u5 x) Hu5Rx) (from left to right) at position 1.
Use reflexivity.
L177064
We prove the intermediate claim Hr5_range: ∀x : set, x ∈ A → apply_fun r5 x ∈ I2.
L177066
We prove the intermediate claim Hu5AB: continuous_map X Tx I0 T0 u5 ∧ (∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third).
(*** show the next residual stays within [-2/3,2/3] on A ***)
L177071
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u5 ∧ (∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r4s ((closed_interval one_third 1) ∩ I) → apply_fun u5 x = one_third) Hu5).
L177077
We prove the intermediate claim Hu5_on_B5: ∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third.
L177081
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u5) (∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third) Hu5AB).
L177085
We prove the intermediate claim Hu5_on_C5: ∀x : set, x ∈ preimage_of A r4s ((closed_interval one_third 1) ∩ I) → apply_fun u5 x = one_third.
L177089
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u5 ∧ (∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r4s ((closed_interval one_third 1) ∩ I) → apply_fun u5 x = one_third) Hu5).
L177095
Let x be given.
L177096
Assume HxA: x ∈ A.
L177096
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L177097
Set I3 to be the term closed_interval one_third 1.
L177098
Set B5 to be the term preimage_of A r4s (I1 ∩ I).
L177099
Set C5 to be the term preimage_of A r4s (I3 ∩ I).
L177100
We prove the intermediate claim Hr4sIx: apply_fun r4s x ∈ I.
L177102
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r4s Hr4s_cont x HxA).
L177102
We prove the intermediate claim HB5_cases: x ∈ B5 ∨ ¬ (x ∈ B5).
L177104
An exact proof term for the current goal is (xm (x ∈ B5)).
L177104
Apply (HB5_cases (apply_fun r5 x ∈ I2)) to the current goal.
L177106
Assume HxB5: x ∈ B5.
L177106
We prove the intermediate claim Hu5eq: apply_fun u5 x = minus_SNo one_third.
L177108
An exact proof term for the current goal is (Hu5_on_B5 x HxB5).
L177108
We prove the intermediate claim Hr5eq: apply_fun r5 x = add_SNo (apply_fun r4s x) one_third.
L177110
rewrite the current goal using (Hr5_apply x HxA) (from left to right).
L177110
rewrite the current goal using Hu5eq (from left to right) at position 1.
L177111
We prove the intermediate claim H13R: one_third ∈ R.
L177113
An exact proof term for the current goal is one_third_in_R.
L177113
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L177115
rewrite the current goal using Hr5eq (from left to right).
L177116
We prove the intermediate claim Hr4sI1I: apply_fun r4s x ∈ I1 ∩ I.
L177118
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r4s x0 ∈ I1 ∩ I) x HxB5).
L177118
We prove the intermediate claim Hr4sI1: apply_fun r4s x ∈ I1.
L177120
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r4s x) Hr4sI1I).
L177120
We prove the intermediate claim H13R: one_third ∈ R.
L177122
An exact proof term for the current goal is one_third_in_R.
L177122
We prove the intermediate claim H23R: two_thirds ∈ R.
L177124
An exact proof term for the current goal is two_thirds_in_R.
L177124
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L177126
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L177126
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L177128
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L177128
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L177130
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L177130
We prove the intermediate claim Hr4sRx: apply_fun r4s x ∈ R.
L177132
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r4s x) Hr4sI1).
L177132
We prove the intermediate claim Hr4s_bounds: Rle (minus_SNo 1) (apply_fun r4s x) ∧ Rle (apply_fun r4s x) (minus_SNo one_third).
L177134
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r4s x) Hm1R Hm13R Hr4sI1).
L177135
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r4s x).
L177137
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r4s x)) (Rle (apply_fun r4s x) (minus_SNo one_third)) Hr4s_bounds).
L177138
We prove the intermediate claim HhiI1: Rle (apply_fun r4s x) (minus_SNo one_third).
L177140
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r4s x)) (Rle (apply_fun r4s x) (minus_SNo one_third)) Hr4s_bounds).
L177141
We prove the intermediate claim Hr5Rx: add_SNo (apply_fun r4s x) one_third ∈ R.
L177143
An exact proof term for the current goal is (real_add_SNo (apply_fun r4s x) Hr4sRx one_third H13R).
L177143
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r4s x) one_third).
L177145
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r4s x) one_third Hm1R Hr4sRx H13R Hm1le).
L177145
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r4s x) one_third).
L177147
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L177147
An exact proof term for the current goal is Hlow_tmp.
L177148
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r4s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L177150
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r4s x) (minus_SNo one_third) one_third Hr4sRx Hm13R H13R HhiI1).
L177150
We prove the intermediate claim H13S: SNo one_third.
L177152
An exact proof term for the current goal is (real_SNo one_third H13R).
L177152
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r4s x) one_third) 0.
L177154
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L177154
An exact proof term for the current goal is Hup0_tmp.
L177155
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L177157
An exact proof term for the current goal is Rle_0_two_thirds.
L177157
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r4s x) one_third) two_thirds.
L177159
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r4s x) one_third) 0 two_thirds Hup0 H0le23).
L177159
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r4s x) one_third) Hm23R H23R Hr5Rx Hlow Hup).
L177163
Assume HxnotB5: ¬ (x ∈ B5).
L177163
We prove the intermediate claim HC5_cases: x ∈ C5 ∨ ¬ (x ∈ C5).
L177165
An exact proof term for the current goal is (xm (x ∈ C5)).
L177165
Apply (HC5_cases (apply_fun r5 x ∈ I2)) to the current goal.
L177167
Assume HxC5: x ∈ C5.
L177167
We prove the intermediate claim Hu5eq: apply_fun u5 x = one_third.
L177169
An exact proof term for the current goal is (Hu5_on_C5 x HxC5).
L177169
We prove the intermediate claim Hr5eq: apply_fun r5 x = add_SNo (apply_fun r4s x) (minus_SNo one_third).
L177171
rewrite the current goal using (Hr5_apply x HxA) (from left to right).
L177171
rewrite the current goal using Hu5eq (from left to right) at position 1.
Use reflexivity.
L177173
rewrite the current goal using Hr5eq (from left to right).
L177174
We prove the intermediate claim Hr4sI3I: apply_fun r4s x ∈ I3 ∩ I.
L177176
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r4s x0 ∈ I3 ∩ I) x HxC5).
L177176
We prove the intermediate claim Hr4sI3: apply_fun r4s x ∈ I3.
L177178
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r4s x) Hr4sI3I).
L177178
We prove the intermediate claim H13R: one_third ∈ R.
L177180
An exact proof term for the current goal is one_third_in_R.
L177180
We prove the intermediate claim H23R: two_thirds ∈ R.
L177182
An exact proof term for the current goal is two_thirds_in_R.
L177182
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L177184
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L177184
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L177186
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L177186
We prove the intermediate claim Hr4sRx: apply_fun r4s x ∈ R.
L177188
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r4s x) Hr4sI3).
L177188
We prove the intermediate claim Hr4s_bounds: Rle one_third (apply_fun r4s x) ∧ Rle (apply_fun r4s x) 1.
L177190
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r4s x) H13R real_1 Hr4sI3).
L177190
We prove the intermediate claim HloI3: Rle one_third (apply_fun r4s x).
L177192
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r4s x)) (Rle (apply_fun r4s x) 1) Hr4s_bounds).
L177192
We prove the intermediate claim HhiI3: Rle (apply_fun r4s x) 1.
L177194
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r4s x)) (Rle (apply_fun r4s x) 1) Hr4s_bounds).
L177194
We prove the intermediate claim Hr5Rx: add_SNo (apply_fun r4s x) (minus_SNo one_third) ∈ R.
L177196
An exact proof term for the current goal is (real_add_SNo (apply_fun r4s x) Hr4sRx (minus_SNo one_third) Hm13R).
L177196
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r4s x) (minus_SNo one_third)).
L177199
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r4s x) (minus_SNo one_third) H13R Hr4sRx Hm13R HloI3).
L177200
We prove the intermediate claim H13S: SNo one_third.
L177202
An exact proof term for the current goal is (real_SNo one_third H13R).
L177202
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r4s x) (minus_SNo one_third)).
L177204
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L177204
An exact proof term for the current goal is H0le_tmp.
L177205
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L177207
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L177207
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r4s x) (minus_SNo one_third)).
L177209
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r4s x) (minus_SNo one_third)) Hm23le0 H0le).
L177211
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r4s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L177214
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r4s x) 1 (minus_SNo one_third) Hr4sRx real_1 Hm13R HhiI3).
L177215
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L177217
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L177217
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L177218
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r4s x) (minus_SNo one_third)) two_thirds.
L177220
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r4s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L177223
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r4s x) (minus_SNo one_third)) Hm23R H23R Hr5Rx Hlow Hup).
L177227
Assume HxnotC5: ¬ (x ∈ C5).
L177227
We prove the intermediate claim HxX: x ∈ X.
L177229
An exact proof term for the current goal is (HAsubX x HxA).
L177229
We prove the intermediate claim HnotI1: ¬ (apply_fun r4s x ∈ I1).
L177231
Assume Hr4sI1': apply_fun r4s x ∈ I1.
L177231
We prove the intermediate claim Hr4sI1I: apply_fun r4s x ∈ I1 ∩ I.
L177233
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r4s x) Hr4sI1' Hr4sIx).
L177233
We prove the intermediate claim HxB5': x ∈ B5.
L177235
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r4s x0 ∈ I1 ∩ I) x HxA Hr4sI1I).
L177235
Apply FalseE to the current goal.
L177236
An exact proof term for the current goal is (HxnotB5 HxB5').
L177237
We prove the intermediate claim HnotI3: ¬ (apply_fun r4s x ∈ I3).
L177239
Assume Hr4sI3': apply_fun r4s x ∈ I3.
L177239
We prove the intermediate claim Hr4sI3I: apply_fun r4s x ∈ I3 ∩ I.
L177241
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r4s x) Hr4sI3' Hr4sIx).
L177241
We prove the intermediate claim HxC5': x ∈ C5.
L177243
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r4s x0 ∈ I3 ∩ I) x HxA Hr4sI3I).
L177243
Apply FalseE to the current goal.
L177244
An exact proof term for the current goal is (HxnotC5 HxC5').
L177245
We prove the intermediate claim Hr4sRx: apply_fun r4s x ∈ R.
L177247
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r4s x) Hr4sIx).
L177247
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L177249
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L177249
We prove the intermediate claim H13R: one_third ∈ R.
L177251
An exact proof term for the current goal is one_third_in_R.
L177251
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L177253
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L177253
We prove the intermediate claim Hr4s_boundsI: Rle (minus_SNo 1) (apply_fun r4s x) ∧ Rle (apply_fun r4s x) 1.
L177255
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r4s x) Hm1R real_1 Hr4sIx).
L177255
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r4s x) (minus_SNo 1)).
L177257
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r4s x) (andEL (Rle (minus_SNo 1) (apply_fun r4s x)) (Rle (apply_fun r4s x) 1) Hr4s_boundsI)).
L177258
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r4s x)).
L177260
An exact proof term for the current goal is (RleE_nlt (apply_fun r4s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r4s x)) (Rle (apply_fun r4s x) 1) Hr4s_boundsI)).
L177261
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r4s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r4s x).
L177263
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r4s x) Hm1R Hm13R Hr4sRx HnotI1).
L177264
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r4s x).
L177266
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r4s x))) to the current goal.
L177267
Assume Hbad: Rlt (apply_fun r4s x) (minus_SNo 1).
L177267
Apply FalseE to the current goal.
L177268
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L177270
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r4s x).
L177270
An exact proof term for the current goal is Hok.
L177271
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r4s x) (minus_SNo one_third)).
L177273
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r4s x) Hm13lt_fx).
L177273
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r4s x) one_third ∨ Rlt 1 (apply_fun r4s x).
L177275
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r4s x) H13R real_1 Hr4sRx HnotI3).
L177276
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r4s x) one_third.
L177278
Apply (HnotI3_cases (Rlt (apply_fun r4s x) one_third)) to the current goal.
L177279
Assume Hok: Rlt (apply_fun r4s x) one_third.
L177279
An exact proof term for the current goal is Hok.
L177281
Assume Hbad: Rlt 1 (apply_fun r4s x).
L177281
Apply FalseE to the current goal.
L177282
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L177283
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r4s x)).
L177285
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r4s x) one_third Hfx_lt_13).
L177285
We prove the intermediate claim Hr4sI0: apply_fun r4s x ∈ I0.
L177287
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L177289
We prove the intermediate claim HxSep: apply_fun r4s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L177291
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r4s x) Hr4sRx (andI (¬ (Rlt (apply_fun r4s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r4s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L177295
rewrite the current goal using HI0_def (from left to right).
L177296
An exact proof term for the current goal is HxSep.
L177297
We prove the intermediate claim Hu5funI0: function_on u5 X I0.
L177299
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u5 Hu5contI0).
L177299
We prove the intermediate claim Hu5xI0: apply_fun u5 x ∈ I0.
L177301
An exact proof term for the current goal is (Hu5funI0 x HxX).
L177301
We prove the intermediate claim Hu5xR: apply_fun u5 x ∈ R.
L177303
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u5 x) Hu5xI0).
L177303
We prove the intermediate claim Hm_u5x_R: minus_SNo (apply_fun u5 x) ∈ R.
L177305
An exact proof term for the current goal is (real_minus_SNo (apply_fun u5 x) Hu5xR).
L177305
rewrite the current goal using (Hr5_apply x HxA) (from left to right).
L177306
We prove the intermediate claim Hr5xR: add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x)) ∈ R.
L177308
An exact proof term for the current goal is (real_add_SNo (apply_fun r4s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r4s x) Hr4sI0) (minus_SNo (apply_fun u5 x)) Hm_u5x_R).
L177310
We prove the intermediate claim H23R: two_thirds ∈ R.
L177312
An exact proof term for the current goal is two_thirds_in_R.
L177312
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L177314
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L177314
We prove the intermediate claim Hr4s_bounds0: Rle (minus_SNo one_third) (apply_fun r4s x) ∧ Rle (apply_fun r4s x) one_third.
L177316
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r4s x) Hm13R H13R Hr4sI0).
L177316
We prove the intermediate claim Hu5_bounds0: Rle (minus_SNo one_third) (apply_fun u5 x) ∧ Rle (apply_fun u5 x) one_third.
L177318
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u5 x) Hm13R H13R Hu5xI0).
L177318
We prove the intermediate claim Hm13_le_r4s: Rle (minus_SNo one_third) (apply_fun r4s x).
L177320
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r4s x)) (Rle (apply_fun r4s x) one_third) Hr4s_bounds0).
L177320
We prove the intermediate claim Hr4s_le_13: Rle (apply_fun r4s x) one_third.
L177322
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r4s x)) (Rle (apply_fun r4s x) one_third) Hr4s_bounds0).
L177322
We prove the intermediate claim Hm13_le_u5x: Rle (minus_SNo one_third) (apply_fun u5 x).
L177324
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u5 x)) (Rle (apply_fun u5 x) one_third) Hu5_bounds0).
L177324
We prove the intermediate claim Hu5x_le_13: Rle (apply_fun u5 x) one_third.
L177326
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u5 x)) (Rle (apply_fun u5 x) one_third) Hu5_bounds0).
L177326
We prove the intermediate claim Hm13_le_mu5: Rle (minus_SNo one_third) (minus_SNo (apply_fun u5 x)).
L177328
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u5 x) one_third Hu5x_le_13).
L177328
We prove the intermediate claim Hmu5_le_13: Rle (minus_SNo (apply_fun u5 x)) one_third.
L177330
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u5 x)) (minus_SNo (minus_SNo one_third)).
L177331
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u5 x) Hm13_le_u5x).
L177331
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L177332
An exact proof term for the current goal is Htmp.
L177333
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u5 x))).
L177336
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u5 x)) Hm13R Hm13R Hm_u5x_R Hm13_le_mu5).
L177337
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u5 x))) (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))).
L177340
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r4s x) (minus_SNo (apply_fun u5 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r4s x) Hr4sI0) Hm_u5x_R Hm13_le_r4s).
L177344
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))).
L177347
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u5 x))) (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))) Hlow1 Hlow2).
L177350
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))).
L177352
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L177352
An exact proof term for the current goal is Hlow_tmp.
L177353
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))) (add_SNo (apply_fun r4s x) one_third).
L177356
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r4s x) (minus_SNo (apply_fun u5 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r4s x) Hr4sI0) Hm_u5x_R H13R Hmu5_le_13).
L177358
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r4s x) one_third) (add_SNo one_third one_third).
L177361
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r4s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r4s x) Hr4sI0) H13R H13R Hr4s_le_13).
L177363
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))) (add_SNo one_third one_third).
L177366
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))) (add_SNo (apply_fun r4s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L177369
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L177371
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))) two_thirds.
L177373
rewrite the current goal using Hdef23 (from left to right) at position 1.
L177373
An exact proof term for the current goal is Hup_tmp.
L177374
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))) Hm23R H23R Hr5xR Hlow Hup).
L177377
Set r5s to be the term compose_fun A r5 (div_const_fun den).
L177378
We prove the intermediate claim Hr5s_cont: continuous_map A Ta I Ti r5s.
L177380
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
L177381
An exact proof term for the current goal is R_standard_topology_is_topology.
L177381
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L177383
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L177383
We prove the intermediate claim Hr5s_contR: continuous_map A Ta R R_standard_topology r5s.
L177385
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r5 (div_const_fun den) Hr5_cont Hdivcont).
L177386
We prove the intermediate claim Hr5s_I: ∀x : set, x ∈ A → apply_fun r5s x ∈ I.
L177388
Let x be given.
L177388
Assume HxA: x ∈ A.
L177388
We prove the intermediate claim Hr5xI2: apply_fun r5 x ∈ I2.
L177390
An exact proof term for the current goal is (Hr5_range x HxA).
L177390
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L177392
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L177392
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L177394
An exact proof term for the current goal is (real_minus_SNo den H23R).
L177394
We prove the intermediate claim Hr5xR: apply_fun r5 x ∈ R.
L177396
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r5 x) Hr5xI2).
L177396
We prove the intermediate claim Hr5xS: SNo (apply_fun r5 x).
L177398
An exact proof term for the current goal is (real_SNo (apply_fun r5 x) Hr5xR).
L177398
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r5 x) ∧ Rle (apply_fun r5 x) den.
L177400
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r5 x) HmdenR H23R Hr5xI2).
L177400
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r5 x).
L177402
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r5 x)) (Rle (apply_fun r5 x) den) Hbounds).
L177404
We prove the intermediate claim Hhi: Rle (apply_fun r5 x) den.
L177406
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r5 x)) (Rle (apply_fun r5 x) den) Hbounds).
L177408
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r5 x)).
L177410
An exact proof term for the current goal is (RleE_nlt (apply_fun r5 x) den Hhi).
L177410
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r5 x) (minus_SNo den)).
L177412
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r5 x) Hlo).
L177412
We prove the intermediate claim HyEq: apply_fun r5s x = div_SNo (apply_fun r5 x) den.
L177414
rewrite the current goal using (compose_fun_apply A r5 (div_const_fun den) x HxA) (from left to right).
L177414
rewrite the current goal using (div_const_fun_apply den (apply_fun r5 x) H23R Hr5xR) (from left to right).
Use reflexivity.
L177416
We prove the intermediate claim HyR: apply_fun r5s x ∈ R.
L177418
rewrite the current goal using HyEq (from left to right).
L177418
An exact proof term for the current goal is (real_div_SNo (apply_fun r5 x) Hr5xR den H23R).
L177419
We prove the intermediate claim HyS: SNo (apply_fun r5s x).
L177421
An exact proof term for the current goal is (real_SNo (apply_fun r5s x) HyR).
L177421
We prove the intermediate claim Hy_le_1: Rle (apply_fun r5s x) 1.
L177423
Apply (RleI (apply_fun r5s x) 1 HyR real_1) to the current goal.
L177423
We will prove ¬ (Rlt 1 (apply_fun r5s x)).
L177424
Assume H1lt: Rlt 1 (apply_fun r5s x).
L177425
We prove the intermediate claim H1lty: 1 < apply_fun r5s x.
L177427
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r5s x) H1lt).
L177427
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r5s x) den.
L177429
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r5s x) den SNo_1 HyS H23S H23pos H1lty).
L177429
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r5s x) den.
L177431
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L177431
An exact proof term for the current goal is HmulLt.
L177432
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r5s x) den = apply_fun r5 x.
L177434
rewrite the current goal using HyEq (from left to right) at position 1.
L177434
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r5 x) den Hr5xS H23S H23ne0).
L177435
We prove the intermediate claim Hden_lt_r5x: den < apply_fun r5 x.
L177437
rewrite the current goal using HmulEq (from right to left).
L177437
An exact proof term for the current goal is HmulLt'.
L177438
We prove the intermediate claim Hbad: Rlt den (apply_fun r5 x).
L177440
An exact proof term for the current goal is (RltI den (apply_fun r5 x) H23R Hr5xR Hden_lt_r5x).
L177440
An exact proof term for the current goal is (Hnlt_hi Hbad).
L177441
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r5s x).
L177443
Apply (RleI (minus_SNo 1) (apply_fun r5s x) Hm1R HyR) to the current goal.
L177443
We will prove ¬ (Rlt (apply_fun r5s x) (minus_SNo 1)).
L177444
Assume Hylt: Rlt (apply_fun r5s x) (minus_SNo 1).
L177445
We prove the intermediate claim Hylts: apply_fun r5s x < minus_SNo 1.
L177447
An exact proof term for the current goal is (RltE_lt (apply_fun r5s x) (minus_SNo 1) Hylt).
L177447
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r5s x) den < mul_SNo (minus_SNo 1) den.
L177449
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r5s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L177450
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r5s x) den = apply_fun r5 x.
L177452
rewrite the current goal using HyEq (from left to right) at position 1.
L177452
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r5 x) den Hr5xS H23S H23ne0).
L177453
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L177455
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L177455
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L177457
We prove the intermediate claim Hr5x_lt_mden: apply_fun r5 x < minus_SNo den.
L177459
rewrite the current goal using HmulEq (from right to left).
L177459
rewrite the current goal using HrhsEq (from right to left).
L177460
An exact proof term for the current goal is HmulLt.
L177461
We prove the intermediate claim Hbad: Rlt (apply_fun r5 x) (minus_SNo den).
L177463
An exact proof term for the current goal is (RltI (apply_fun r5 x) (minus_SNo den) Hr5xR HmdenR Hr5x_lt_mden).
L177463
An exact proof term for the current goal is (Hnlt_lo Hbad).
L177464
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r5s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L177466
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r5s I Hr5s_contR HIcR Hr5s_I).
L177467
We prove the intermediate claim Hex_u6: ∃u6 : set, continuous_map X Tx I0 T0 u6 ∧ (∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r5s ((closed_interval one_third 1) ∩ I) → apply_fun u6 x = one_third).
(*** ninth correction step scaffold: build u6 and g6 from residual r5s ***)
L177477
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r5s Hnorm HA Hr5s_cont).
L177477
Apply Hex_u6 to the current goal.
L177478
Let u6 be given.
L177479
Assume Hu6.
L177479
We prove the intermediate claim Hu6contI0: continuous_map X Tx I0 T0 u6.
L177481
We prove the intermediate claim Hu6AB: continuous_map X Tx I0 T0 u6 ∧ (∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third).
L177485
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u6 ∧ (∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r5s ((closed_interval one_third 1) ∩ I) → apply_fun u6 x = one_third) Hu6).
L177491
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u6) (∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third) Hu6AB).
L177496
We prove the intermediate claim Hu6contR: continuous_map X Tx R R_standard_topology u6.
L177498
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L177499
We prove the intermediate claim HI0subR: I0 ⊆ R.
L177501
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L177501
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u6 Hu6contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L177509
Set den6 to be the term mul_SNo den5 den.
L177510
We prove the intermediate claim Hden6R: den6 ∈ R.
L177512
An exact proof term for the current goal is (real_mul_SNo den5 Hden5R den H23R).
L177512
We prove the intermediate claim Hden6pos: 0 < den6.
L177514
We prove the intermediate claim Hden5S: SNo den5.
L177515
An exact proof term for the current goal is (real_SNo den5 Hden5R).
L177515
An exact proof term for the current goal is (mul_SNo_pos_pos den5 den Hden5S H23S Hden5pos HdenPos).
L177516
Set u6s to be the term compose_fun X u6 (mul_const_fun den6).
L177517
We prove the intermediate claim Hu6s_cont: continuous_map X Tx R R_standard_topology u6s.
L177519
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u6 den6 HTx Hu6contR Hden6R Hden6pos).
L177519
Set g6 to be the term compose_fun X (pair_map X g5 u6s) add_fun_R.
L177520
We prove the intermediate claim Hg6cont: continuous_map X Tx R R_standard_topology g6.
L177522
An exact proof term for the current goal is (add_two_continuous_R X Tx g5 u6s HTx Hg5cont Hu6s_cont).
L177522
We prove the intermediate claim Hu6contA: continuous_map A Ta R R_standard_topology u6.
(*** tenth correction step scaffold: residual r6 on A and scaling r6s ***)
L177526
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u6 A HTx HAsubX Hu6contR).
L177526
Set u6neg to be the term compose_fun A u6 neg_fun.
L177527
We prove the intermediate claim Hu6neg_cont: continuous_map A Ta R R_standard_topology u6neg.
L177529
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u6 neg_fun Hu6contA Hnegcont).
L177530
We prove the intermediate claim Hr5s_contR: continuous_map A Ta R R_standard_topology r5s.
L177532
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r5s Hr5s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L177533
Set r6 to be the term compose_fun A (pair_map A r5s u6neg) add_fun_R.
L177534
We prove the intermediate claim Hr6_cont: continuous_map A Ta R R_standard_topology r6.
L177536
An exact proof term for the current goal is (add_two_continuous_R A Ta r5s u6neg HTa Hr5s_contR Hu6neg_cont).
L177536
We prove the intermediate claim Hr6_apply: ∀x : set, x ∈ A → apply_fun r6 x = add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x)).
L177539
Let x be given.
L177539
Assume HxA: x ∈ A.
L177539
We prove the intermediate claim Hpimg: apply_fun (pair_map A r5s u6neg) x ∈ setprod R R.
L177541
rewrite the current goal using (pair_map_apply A R R r5s u6neg x HxA) (from left to right).
L177541
We prove the intermediate claim Hr5sxI: apply_fun r5s x ∈ I.
L177543
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r5s Hr5s_cont x HxA).
L177543
We prove the intermediate claim Hr5sxR: apply_fun r5s x ∈ R.
L177545
An exact proof term for the current goal is (HIcR (apply_fun r5s x) Hr5sxI).
L177545
We prove the intermediate claim Hu6negRx: apply_fun u6neg x ∈ R.
L177547
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u6neg Hu6neg_cont x HxA).
L177547
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r5s x) (apply_fun u6neg x) Hr5sxR Hu6negRx).
L177548
rewrite the current goal using (compose_fun_apply A (pair_map A r5s u6neg) add_fun_R x HxA) (from left to right).
L177549
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r5s u6neg) x) Hpimg) (from left to right) at position 1.
L177550
rewrite the current goal using (pair_map_apply A R R r5s u6neg x HxA) (from left to right).
L177551
rewrite the current goal using (tuple_2_0_eq (apply_fun r5s x) (apply_fun u6neg x)) (from left to right).
L177552
rewrite the current goal using (tuple_2_1_eq (apply_fun r5s x) (apply_fun u6neg x)) (from left to right).
L177553
rewrite the current goal using (compose_fun_apply A u6 neg_fun x HxA) (from left to right) at position 1.
L177554
We prove the intermediate claim Hu6Rx: apply_fun u6 x ∈ R.
L177556
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u6 Hu6contA x HxA).
L177556
rewrite the current goal using (neg_fun_apply (apply_fun u6 x) Hu6Rx) (from left to right) at position 1.
Use reflexivity.
L177558
We prove the intermediate claim Hr6_range: ∀x : set, x ∈ A → apply_fun r6 x ∈ I2.
L177560
We prove the intermediate claim Hu6AB: continuous_map X Tx I0 T0 u6 ∧ (∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third).
(*** show the next residual stays within [-2/3,2/3] on A ***)
L177565
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u6 ∧ (∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r5s ((closed_interval one_third 1) ∩ I) → apply_fun u6 x = one_third) Hu6).
L177571
We prove the intermediate claim Hu6_on_B6: ∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third.
L177575
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u6) (∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third) Hu6AB).
L177579
We prove the intermediate claim Hu6_on_C6: ∀x : set, x ∈ preimage_of A r5s ((closed_interval one_third 1) ∩ I) → apply_fun u6 x = one_third.
L177583
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u6 ∧ (∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r5s ((closed_interval one_third 1) ∩ I) → apply_fun u6 x = one_third) Hu6).
L177589
Let x be given.
L177590
Assume HxA: x ∈ A.
L177590
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L177591
Set I3 to be the term closed_interval one_third 1.
L177592
Set B6 to be the term preimage_of A r5s (I1 ∩ I).
L177593
Set C6 to be the term preimage_of A r5s (I3 ∩ I).
L177594
We prove the intermediate claim Hr5sIx: apply_fun r5s x ∈ I.
L177596
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r5s Hr5s_cont x HxA).
L177596
We prove the intermediate claim HB6_cases: x ∈ B6 ∨ ¬ (x ∈ B6).
L177598
An exact proof term for the current goal is (xm (x ∈ B6)).
L177598
Apply (HB6_cases (apply_fun r6 x ∈ I2)) to the current goal.
L177600
Assume HxB6: x ∈ B6.
L177600
We prove the intermediate claim Hu6eq: apply_fun u6 x = minus_SNo one_third.
L177602
An exact proof term for the current goal is (Hu6_on_B6 x HxB6).
L177602
We prove the intermediate claim Hr6eq: apply_fun r6 x = add_SNo (apply_fun r5s x) one_third.
L177604
rewrite the current goal using (Hr6_apply x HxA) (from left to right).
L177604
rewrite the current goal using Hu6eq (from left to right) at position 1.
L177605
We prove the intermediate claim H13R: one_third ∈ R.
L177607
An exact proof term for the current goal is one_third_in_R.
L177607
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L177609
rewrite the current goal using Hr6eq (from left to right).
L177610
We prove the intermediate claim Hr5sI1I: apply_fun r5s x ∈ I1 ∩ I.
L177612
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r5s x0 ∈ I1 ∩ I) x HxB6).
L177612
We prove the intermediate claim Hr5sI1: apply_fun r5s x ∈ I1.
L177614
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r5s x) Hr5sI1I).
L177614
We prove the intermediate claim H13R: one_third ∈ R.
L177616
An exact proof term for the current goal is one_third_in_R.
L177616
We prove the intermediate claim H23R: two_thirds ∈ R.
L177618
An exact proof term for the current goal is two_thirds_in_R.
L177618
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L177620
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L177620
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L177622
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L177622
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L177624
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L177624
We prove the intermediate claim Hr5sRx: apply_fun r5s x ∈ R.
L177626
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r5s x) Hr5sI1).
L177626
We prove the intermediate claim Hr5s_bounds: Rle (minus_SNo 1) (apply_fun r5s x) ∧ Rle (apply_fun r5s x) (minus_SNo one_third).
L177628
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r5s x) Hm1R Hm13R Hr5sI1).
L177629
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r5s x).
L177631
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r5s x)) (Rle (apply_fun r5s x) (minus_SNo one_third)) Hr5s_bounds).
L177632
We prove the intermediate claim HhiI1: Rle (apply_fun r5s x) (minus_SNo one_third).
L177634
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r5s x)) (Rle (apply_fun r5s x) (minus_SNo one_third)) Hr5s_bounds).
L177635
We prove the intermediate claim Hr6Rx: add_SNo (apply_fun r5s x) one_third ∈ R.
L177637
An exact proof term for the current goal is (real_add_SNo (apply_fun r5s x) Hr5sRx one_third H13R).
L177637
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r5s x) one_third).
L177639
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r5s x) one_third Hm1R Hr5sRx H13R Hm1le).
L177639
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r5s x) one_third).
L177641
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L177641
An exact proof term for the current goal is Hlow_tmp.
L177642
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r5s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L177644
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r5s x) (minus_SNo one_third) one_third Hr5sRx Hm13R H13R HhiI1).
L177644
We prove the intermediate claim H13S: SNo one_third.
L177646
An exact proof term for the current goal is (real_SNo one_third H13R).
L177646
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r5s x) one_third) 0.
L177648
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L177648
An exact proof term for the current goal is Hup0_tmp.
L177649
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L177651
An exact proof term for the current goal is Rle_0_two_thirds.
L177651
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r5s x) one_third) two_thirds.
L177653
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r5s x) one_third) 0 two_thirds Hup0 H0le23).
L177653
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r5s x) one_third) Hm23R H23R Hr6Rx Hlow Hup).
L177657
Assume HxnotB6: ¬ (x ∈ B6).
L177657
We prove the intermediate claim HC6_cases: x ∈ C6 ∨ ¬ (x ∈ C6).
L177659
An exact proof term for the current goal is (xm (x ∈ C6)).
L177659
Apply (HC6_cases (apply_fun r6 x ∈ I2)) to the current goal.
L177661
Assume HxC6: x ∈ C6.
L177661
We prove the intermediate claim Hu6eq: apply_fun u6 x = one_third.
L177663
An exact proof term for the current goal is (Hu6_on_C6 x HxC6).
L177663
We prove the intermediate claim Hr6eq: apply_fun r6 x = add_SNo (apply_fun r5s x) (minus_SNo one_third).
L177665
rewrite the current goal using (Hr6_apply x HxA) (from left to right).
L177665
rewrite the current goal using Hu6eq (from left to right) at position 1.
Use reflexivity.
L177667
rewrite the current goal using Hr6eq (from left to right).
L177668
We prove the intermediate claim Hr5sI3I: apply_fun r5s x ∈ I3 ∩ I.
L177670
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r5s x0 ∈ I3 ∩ I) x HxC6).
L177670
We prove the intermediate claim Hr5sI3: apply_fun r5s x ∈ I3.
L177672
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r5s x) Hr5sI3I).
L177672
We prove the intermediate claim H13R: one_third ∈ R.
L177674
An exact proof term for the current goal is one_third_in_R.
L177674
We prove the intermediate claim H23R: two_thirds ∈ R.
L177676
An exact proof term for the current goal is two_thirds_in_R.
L177676
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L177678
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L177678
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L177680
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L177680
We prove the intermediate claim Hr5sRx: apply_fun r5s x ∈ R.
L177682
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r5s x) Hr5sI3).
L177682
We prove the intermediate claim Hr5s_bounds: Rle one_third (apply_fun r5s x) ∧ Rle (apply_fun r5s x) 1.
L177684
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r5s x) H13R real_1 Hr5sI3).
L177684
We prove the intermediate claim HloI3: Rle one_third (apply_fun r5s x).
L177686
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r5s x)) (Rle (apply_fun r5s x) 1) Hr5s_bounds).
L177686
We prove the intermediate claim HhiI3: Rle (apply_fun r5s x) 1.
L177688
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r5s x)) (Rle (apply_fun r5s x) 1) Hr5s_bounds).
L177688
We prove the intermediate claim Hr6Rx: add_SNo (apply_fun r5s x) (minus_SNo one_third) ∈ R.
L177690
An exact proof term for the current goal is (real_add_SNo (apply_fun r5s x) Hr5sRx (minus_SNo one_third) Hm13R).
L177690
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r5s x) (minus_SNo one_third)).
L177693
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r5s x) (minus_SNo one_third) H13R Hr5sRx Hm13R HloI3).
L177694
We prove the intermediate claim H13S: SNo one_third.
L177696
An exact proof term for the current goal is (real_SNo one_third H13R).
L177696
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r5s x) (minus_SNo one_third)).
L177698
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L177698
An exact proof term for the current goal is H0le_tmp.
L177699
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L177701
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L177701
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r5s x) (minus_SNo one_third)).
L177703
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r5s x) (minus_SNo one_third)) Hm23le0 H0le).
L177705
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r5s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L177708
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r5s x) 1 (minus_SNo one_third) Hr5sRx real_1 Hm13R HhiI3).
L177709
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L177711
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L177711
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L177712
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r5s x) (minus_SNo one_third)) two_thirds.
L177714
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r5s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L177717
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r5s x) (minus_SNo one_third)) Hm23R H23R Hr6Rx Hlow Hup).
L177721
Assume HxnotC6: ¬ (x ∈ C6).
L177721
We prove the intermediate claim HxX: x ∈ X.
L177723
An exact proof term for the current goal is (HAsubX x HxA).
L177723
We prove the intermediate claim HnotI1: ¬ (apply_fun r5s x ∈ I1).
L177725
Assume Hr5sI1': apply_fun r5s x ∈ I1.
L177725
We prove the intermediate claim Hr5sI1I: apply_fun r5s x ∈ I1 ∩ I.
L177727
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r5s x) Hr5sI1' Hr5sIx).
L177727
We prove the intermediate claim HxB6': x ∈ B6.
L177729
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r5s x0 ∈ I1 ∩ I) x HxA Hr5sI1I).
L177729
Apply FalseE to the current goal.
L177730
An exact proof term for the current goal is (HxnotB6 HxB6').
L177731
We prove the intermediate claim HnotI3: ¬ (apply_fun r5s x ∈ I3).
L177733
Assume Hr5sI3': apply_fun r5s x ∈ I3.
L177733
We prove the intermediate claim Hr5sI3I: apply_fun r5s x ∈ I3 ∩ I.
L177735
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r5s x) Hr5sI3' Hr5sIx).
L177735
We prove the intermediate claim HxC6': x ∈ C6.
L177737
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r5s x0 ∈ I3 ∩ I) x HxA Hr5sI3I).
L177737
Apply FalseE to the current goal.
L177738
An exact proof term for the current goal is (HxnotC6 HxC6').
L177739
We prove the intermediate claim Hr5sRx: apply_fun r5s x ∈ R.
L177741
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r5s x) Hr5sIx).
L177741
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L177743
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L177743
We prove the intermediate claim H13R: one_third ∈ R.
L177745
An exact proof term for the current goal is one_third_in_R.
L177745
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L177747
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L177747
We prove the intermediate claim Hr5s_boundsI: Rle (minus_SNo 1) (apply_fun r5s x) ∧ Rle (apply_fun r5s x) 1.
L177749
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r5s x) Hm1R real_1 Hr5sIx).
L177749
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r5s x) (minus_SNo 1)).
L177751
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r5s x) (andEL (Rle (minus_SNo 1) (apply_fun r5s x)) (Rle (apply_fun r5s x) 1) Hr5s_boundsI)).
L177752
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r5s x)).
L177754
An exact proof term for the current goal is (RleE_nlt (apply_fun r5s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r5s x)) (Rle (apply_fun r5s x) 1) Hr5s_boundsI)).
L177755
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r5s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r5s x).
L177757
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r5s x) Hm1R Hm13R Hr5sRx HnotI1).
L177758
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r5s x).
L177760
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r5s x))) to the current goal.
L177761
Assume Hbad: Rlt (apply_fun r5s x) (minus_SNo 1).
L177761
Apply FalseE to the current goal.
L177762
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L177764
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r5s x).
L177764
An exact proof term for the current goal is Hok.
L177765
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r5s x) (minus_SNo one_third)).
L177767
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r5s x) Hm13lt_fx).
L177767
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r5s x) one_third ∨ Rlt 1 (apply_fun r5s x).
L177769
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r5s x) H13R real_1 Hr5sRx HnotI3).
L177770
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r5s x) one_third.
L177772
Apply (HnotI3_cases (Rlt (apply_fun r5s x) one_third)) to the current goal.
L177773
Assume Hok: Rlt (apply_fun r5s x) one_third.
L177773
An exact proof term for the current goal is Hok.
L177775
Assume Hbad: Rlt 1 (apply_fun r5s x).
L177775
Apply FalseE to the current goal.
L177776
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L177777
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r5s x)).
L177779
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r5s x) one_third Hfx_lt_13).
L177779
We prove the intermediate claim Hr5sI0: apply_fun r5s x ∈ I0.
L177781
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L177783
We prove the intermediate claim HxSep: apply_fun r5s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L177785
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r5s x) Hr5sRx (andI (¬ (Rlt (apply_fun r5s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r5s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L177789
rewrite the current goal using HI0_def (from left to right).
L177790
An exact proof term for the current goal is HxSep.
L177791
We prove the intermediate claim Hu6funI0: function_on u6 X I0.
L177793
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u6 Hu6contI0).
L177793
We prove the intermediate claim Hu6xI0: apply_fun u6 x ∈ I0.
L177795
An exact proof term for the current goal is (Hu6funI0 x HxX).
L177795
We prove the intermediate claim Hu6xR: apply_fun u6 x ∈ R.
L177797
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u6 x) Hu6xI0).
L177797
We prove the intermediate claim Hm_u6x_R: minus_SNo (apply_fun u6 x) ∈ R.
L177799
An exact proof term for the current goal is (real_minus_SNo (apply_fun u6 x) Hu6xR).
L177799
rewrite the current goal using (Hr6_apply x HxA) (from left to right).
L177800
We prove the intermediate claim Hr6xR: add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x)) ∈ R.
L177802
An exact proof term for the current goal is (real_add_SNo (apply_fun r5s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r5s x) Hr5sI0) (minus_SNo (apply_fun u6 x)) Hm_u6x_R).
L177804
We prove the intermediate claim H23R: two_thirds ∈ R.
L177806
An exact proof term for the current goal is two_thirds_in_R.
L177806
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L177808
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L177808
We prove the intermediate claim Hr5s_bounds0: Rle (minus_SNo one_third) (apply_fun r5s x) ∧ Rle (apply_fun r5s x) one_third.
L177810
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r5s x) Hm13R H13R Hr5sI0).
L177810
We prove the intermediate claim Hu6_bounds0: Rle (minus_SNo one_third) (apply_fun u6 x) ∧ Rle (apply_fun u6 x) one_third.
L177812
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u6 x) Hm13R H13R Hu6xI0).
L177812
We prove the intermediate claim Hm13_le_r5s: Rle (minus_SNo one_third) (apply_fun r5s x).
L177814
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r5s x)) (Rle (apply_fun r5s x) one_third) Hr5s_bounds0).
L177814
We prove the intermediate claim Hr5s_le_13: Rle (apply_fun r5s x) one_third.
L177816
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r5s x)) (Rle (apply_fun r5s x) one_third) Hr5s_bounds0).
L177816
We prove the intermediate claim Hm13_le_u6x: Rle (minus_SNo one_third) (apply_fun u6 x).
L177818
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u6 x)) (Rle (apply_fun u6 x) one_third) Hu6_bounds0).
L177818
We prove the intermediate claim Hu6x_le_13: Rle (apply_fun u6 x) one_third.
L177820
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u6 x)) (Rle (apply_fun u6 x) one_third) Hu6_bounds0).
L177820
We prove the intermediate claim Hm13_le_mu6: Rle (minus_SNo one_third) (minus_SNo (apply_fun u6 x)).
L177822
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u6 x) one_third Hu6x_le_13).
L177822
We prove the intermediate claim Hmu6_le_13: Rle (minus_SNo (apply_fun u6 x)) one_third.
L177824
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u6 x)) (minus_SNo (minus_SNo one_third)).
L177825
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u6 x) Hm13_le_u6x).
L177825
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L177826
An exact proof term for the current goal is Htmp.
L177827
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u6 x))).
L177830
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u6 x)) Hm13R Hm13R Hm_u6x_R Hm13_le_mu6).
L177831
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u6 x))) (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))).
L177834
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r5s x) (minus_SNo (apply_fun u6 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r5s x) Hr5sI0) Hm_u6x_R Hm13_le_r5s).
L177838
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))).
L177841
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u6 x))) (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))) Hlow1 Hlow2).
L177844
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))).
L177846
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L177846
An exact proof term for the current goal is Hlow_tmp.
L177847
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))) (add_SNo (apply_fun r5s x) one_third).
L177850
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r5s x) (minus_SNo (apply_fun u6 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r5s x) Hr5sI0) Hm_u6x_R H13R Hmu6_le_13).
L177852
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r5s x) one_third) (add_SNo one_third one_third).
L177855
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r5s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r5s x) Hr5sI0) H13R H13R Hr5s_le_13).
L177857
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))) (add_SNo one_third one_third).
L177860
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))) (add_SNo (apply_fun r5s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L177863
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L177865
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))) two_thirds.
L177867
rewrite the current goal using Hdef23 (from left to right) at position 1.
L177867
An exact proof term for the current goal is Hup_tmp.
L177868
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))) Hm23R H23R Hr6xR Hlow Hup).
L177871
Set r6s to be the term compose_fun A r6 (div_const_fun den).
L177872
We prove the intermediate claim Hr6s_cont: continuous_map A Ta I Ti r6s.
L177874
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
L177875
An exact proof term for the current goal is R_standard_topology_is_topology.
L177875
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L177877
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L177877
We prove the intermediate claim Hr6s_contR: continuous_map A Ta R R_standard_topology r6s.
L177879
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r6 (div_const_fun den) Hr6_cont Hdivcont).
L177880
We prove the intermediate claim Hr6s_I: ∀x : set, x ∈ A → apply_fun r6s x ∈ I.
L177882
Let x be given.
L177882
Assume HxA: x ∈ A.
L177882
We prove the intermediate claim Hr6xI2: apply_fun r6 x ∈ I2.
L177884
An exact proof term for the current goal is (Hr6_range x HxA).
L177884
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L177886
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L177886
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L177888
An exact proof term for the current goal is (real_minus_SNo den H23R).
L177888
We prove the intermediate claim Hr6xR: apply_fun r6 x ∈ R.
L177890
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r6 x) Hr6xI2).
L177890
We prove the intermediate claim Hr6xS: SNo (apply_fun r6 x).
L177892
An exact proof term for the current goal is (real_SNo (apply_fun r6 x) Hr6xR).
L177892
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r6 x) ∧ Rle (apply_fun r6 x) den.
L177894
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r6 x) HmdenR H23R Hr6xI2).
L177894
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r6 x).
L177896
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r6 x)) (Rle (apply_fun r6 x) den) Hbounds).
L177898
We prove the intermediate claim Hhi: Rle (apply_fun r6 x) den.
L177900
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r6 x)) (Rle (apply_fun r6 x) den) Hbounds).
L177902
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r6 x)).
L177904
An exact proof term for the current goal is (RleE_nlt (apply_fun r6 x) den Hhi).
L177904
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r6 x) (minus_SNo den)).
L177906
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r6 x) Hlo).
L177906
We prove the intermediate claim HyEq: apply_fun r6s x = div_SNo (apply_fun r6 x) den.
L177908
rewrite the current goal using (compose_fun_apply A r6 (div_const_fun den) x HxA) (from left to right).
L177908
rewrite the current goal using (div_const_fun_apply den (apply_fun r6 x) H23R Hr6xR) (from left to right).
Use reflexivity.
L177910
We prove the intermediate claim HyR: apply_fun r6s x ∈ R.
L177912
rewrite the current goal using HyEq (from left to right).
L177912
An exact proof term for the current goal is (real_div_SNo (apply_fun r6 x) Hr6xR den H23R).
L177913
We prove the intermediate claim HyS: SNo (apply_fun r6s x).
L177915
An exact proof term for the current goal is (real_SNo (apply_fun r6s x) HyR).
L177915
We prove the intermediate claim Hy_le_1: Rle (apply_fun r6s x) 1.
L177917
Apply (RleI (apply_fun r6s x) 1 HyR real_1) to the current goal.
L177917
We will prove ¬ (Rlt 1 (apply_fun r6s x)).
L177918
Assume H1lt: Rlt 1 (apply_fun r6s x).
L177919
We prove the intermediate claim H1lty: 1 < apply_fun r6s x.
L177921
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r6s x) H1lt).
L177921
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r6s x) den.
L177923
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r6s x) den SNo_1 HyS H23S H23pos H1lty).
L177923
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r6s x) den.
L177925
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L177925
An exact proof term for the current goal is HmulLt.
L177926
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r6s x) den = apply_fun r6 x.
L177928
rewrite the current goal using HyEq (from left to right) at position 1.
L177928
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r6 x) den Hr6xS H23S H23ne0).
L177929
We prove the intermediate claim Hden_lt_r6x: den < apply_fun r6 x.
L177931
rewrite the current goal using HmulEq (from right to left).
L177931
An exact proof term for the current goal is HmulLt'.
L177932
We prove the intermediate claim Hbad: Rlt den (apply_fun r6 x).
L177934
An exact proof term for the current goal is (RltI den (apply_fun r6 x) H23R Hr6xR Hden_lt_r6x).
L177934
An exact proof term for the current goal is (Hnlt_hi Hbad).
L177935
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r6s x).
L177937
Apply (RleI (minus_SNo 1) (apply_fun r6s x) Hm1R HyR) to the current goal.
L177937
We will prove ¬ (Rlt (apply_fun r6s x) (minus_SNo 1)).
L177938
Assume Hylt: Rlt (apply_fun r6s x) (minus_SNo 1).
L177939
We prove the intermediate claim Hylts: apply_fun r6s x < minus_SNo 1.
L177941
An exact proof term for the current goal is (RltE_lt (apply_fun r6s x) (minus_SNo 1) Hylt).
L177941
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r6s x) den < mul_SNo (minus_SNo 1) den.
L177943
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r6s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L177944
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r6s x) den = apply_fun r6 x.
L177946
rewrite the current goal using HyEq (from left to right) at position 1.
L177946
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r6 x) den Hr6xS H23S H23ne0).
L177947
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L177949
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L177949
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L177951
We prove the intermediate claim Hr6x_lt_mden: apply_fun r6 x < minus_SNo den.
L177953
rewrite the current goal using HmulEq (from right to left).
L177953
rewrite the current goal using HrhsEq (from right to left).
L177954
An exact proof term for the current goal is HmulLt.
L177955
We prove the intermediate claim Hbad: Rlt (apply_fun r6 x) (minus_SNo den).
L177957
An exact proof term for the current goal is (RltI (apply_fun r6 x) (minus_SNo den) Hr6xR HmdenR Hr6x_lt_mden).
L177957
An exact proof term for the current goal is (Hnlt_lo Hbad).
L177958
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r6s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L177960
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r6s I Hr6s_contR HIcR Hr6s_I).
L177961
We prove the intermediate claim Hex_u7: ∃u7 : set, continuous_map X Tx I0 T0 u7 ∧ (∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r6s ((closed_interval one_third 1) ∩ I) → apply_fun u7 x = one_third).
(*** eleventh correction step scaffold: build u7 and g7 from residual r6s ***)
L177971
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r6s Hnorm HA Hr6s_cont).
L177971
Apply Hex_u7 to the current goal.
L177972
Let u7 be given.
L177973
Assume Hu7.
L177973
We prove the intermediate claim Hu7contI0: continuous_map X Tx I0 T0 u7.
L177975
We prove the intermediate claim Hu7AB: continuous_map X Tx I0 T0 u7 ∧ (∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third).
L177979
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u7 ∧ (∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r6s ((closed_interval one_third 1) ∩ I) → apply_fun u7 x = one_third) Hu7).
L177985
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u7) (∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third) Hu7AB).
L177990
We prove the intermediate claim Hu7contR: continuous_map X Tx R R_standard_topology u7.
L177992
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L177993
We prove the intermediate claim HI0subR: I0 ⊆ R.
L177995
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L177995
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u7 Hu7contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L178003
Set den7 to be the term mul_SNo den6 den.
L178004
We prove the intermediate claim Hden7R: den7 ∈ R.
L178006
An exact proof term for the current goal is (real_mul_SNo den6 Hden6R den H23R).
L178006
We prove the intermediate claim Hden7pos: 0 < den7.
L178008
We prove the intermediate claim Hden6S: SNo den6.
L178009
An exact proof term for the current goal is (real_SNo den6 Hden6R).
L178009
An exact proof term for the current goal is (mul_SNo_pos_pos den6 den Hden6S H23S Hden6pos HdenPos).
L178010
Set u7s to be the term compose_fun X u7 (mul_const_fun den7).
L178011
We prove the intermediate claim Hu7s_cont: continuous_map X Tx R R_standard_topology u7s.
L178013
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u7 den7 HTx Hu7contR Hden7R Hden7pos).
L178013
Set g7 to be the term compose_fun X (pair_map X g6 u7s) add_fun_R.
L178014
We prove the intermediate claim Hg7cont: continuous_map X Tx R R_standard_topology g7.
L178016
An exact proof term for the current goal is (add_two_continuous_R X Tx g6 u7s HTx Hg6cont Hu7s_cont).
L178016
We prove the intermediate claim Hu7contA: continuous_map A Ta R R_standard_topology u7.
(*** twelfth correction step scaffold: residual r7 on A and scaling r7s ***)
L178020
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u7 A HTx HAsubX Hu7contR).
L178020
Set u7neg to be the term compose_fun A u7 neg_fun.
L178021
We prove the intermediate claim Hu7neg_cont: continuous_map A Ta R R_standard_topology u7neg.
L178023
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u7 neg_fun Hu7contA Hnegcont).
L178024
We prove the intermediate claim Hr6s_contR: continuous_map A Ta R R_standard_topology r6s.
L178026
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r6s Hr6s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L178027
Set r7 to be the term compose_fun A (pair_map A r6s u7neg) add_fun_R.
L178028
We prove the intermediate claim Hr7_cont: continuous_map A Ta R R_standard_topology r7.
L178030
An exact proof term for the current goal is (add_two_continuous_R A Ta r6s u7neg HTa Hr6s_contR Hu7neg_cont).
L178030
We prove the intermediate claim Hr7_apply: ∀x : set, x ∈ A → apply_fun r7 x = add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x)).
L178033
Let x be given.
L178033
Assume HxA: x ∈ A.
L178033
We prove the intermediate claim Hpimg: apply_fun (pair_map A r6s u7neg) x ∈ setprod R R.
L178035
rewrite the current goal using (pair_map_apply A R R r6s u7neg x HxA) (from left to right).
L178035
We prove the intermediate claim Hr6sxI: apply_fun r6s x ∈ I.
L178037
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r6s Hr6s_cont x HxA).
L178037
We prove the intermediate claim Hr6sxR: apply_fun r6s x ∈ R.
L178039
An exact proof term for the current goal is (HIcR (apply_fun r6s x) Hr6sxI).
L178039
We prove the intermediate claim Hu7negRx: apply_fun u7neg x ∈ R.
L178041
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u7neg Hu7neg_cont x HxA).
L178041
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r6s x) (apply_fun u7neg x) Hr6sxR Hu7negRx).
L178042
rewrite the current goal using (compose_fun_apply A (pair_map A r6s u7neg) add_fun_R x HxA) (from left to right).
L178043
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r6s u7neg) x) Hpimg) (from left to right) at position 1.
L178044
rewrite the current goal using (pair_map_apply A R R r6s u7neg x HxA) (from left to right).
L178045
rewrite the current goal using (tuple_2_0_eq (apply_fun r6s x) (apply_fun u7neg x)) (from left to right).
L178046
rewrite the current goal using (tuple_2_1_eq (apply_fun r6s x) (apply_fun u7neg x)) (from left to right).
L178047
rewrite the current goal using (compose_fun_apply A u7 neg_fun x HxA) (from left to right) at position 1.
L178048
We prove the intermediate claim Hu7Rx: apply_fun u7 x ∈ R.
L178050
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u7 Hu7contA x HxA).
L178050
rewrite the current goal using (neg_fun_apply (apply_fun u7 x) Hu7Rx) (from left to right) at position 1.
Use reflexivity.
L178052
We prove the intermediate claim Hr7_range: ∀x : set, x ∈ A → apply_fun r7 x ∈ I2.
L178054
We prove the intermediate claim Hu7AB: continuous_map X Tx I0 T0 u7 ∧ (∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third).
L178058
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u7 ∧ (∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r6s ((closed_interval one_third 1) ∩ I) → apply_fun u7 x = one_third) Hu7).
L178064
We prove the intermediate claim Hu7_on_B7: ∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third.
L178068
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u7) (∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third) Hu7AB).
L178072
We prove the intermediate claim Hu7_on_C7: ∀x : set, x ∈ preimage_of A r6s ((closed_interval one_third 1) ∩ I) → apply_fun u7 x = one_third.
L178076
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u7 ∧ (∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r6s ((closed_interval one_third 1) ∩ I) → apply_fun u7 x = one_third) Hu7).
L178082
Let x be given.
L178083
Assume HxA: x ∈ A.
L178083
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L178084
Set I3 to be the term closed_interval one_third 1.
L178085
Set B7 to be the term preimage_of A r6s (I1 ∩ I).
L178086
Set C7 to be the term preimage_of A r6s (I3 ∩ I).
L178087
We prove the intermediate claim Hr6sIx: apply_fun r6s x ∈ I.
L178089
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r6s Hr6s_cont x HxA).
L178089
We prove the intermediate claim HB7_cases: x ∈ B7 ∨ ¬ (x ∈ B7).
L178091
An exact proof term for the current goal is (xm (x ∈ B7)).
L178091
Apply (HB7_cases (apply_fun r7 x ∈ I2)) to the current goal.
L178093
Assume HxB7: x ∈ B7.
L178093
We prove the intermediate claim Hu7eq: apply_fun u7 x = minus_SNo one_third.
L178095
An exact proof term for the current goal is (Hu7_on_B7 x HxB7).
L178095
We prove the intermediate claim Hr7eq: apply_fun r7 x = add_SNo (apply_fun r6s x) one_third.
L178097
rewrite the current goal using (Hr7_apply x HxA) (from left to right).
L178097
rewrite the current goal using Hu7eq (from left to right) at position 1.
L178098
We prove the intermediate claim H13R: one_third ∈ R.
L178100
An exact proof term for the current goal is one_third_in_R.
L178100
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L178102
rewrite the current goal using Hr7eq (from left to right).
L178103
We prove the intermediate claim Hr6sI1I: apply_fun r6s x ∈ I1 ∩ I.
L178105
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r6s x0 ∈ I1 ∩ I) x HxB7).
L178105
We prove the intermediate claim Hr6sI1: apply_fun r6s x ∈ I1.
L178107
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r6s x) Hr6sI1I).
L178107
We prove the intermediate claim H13R: one_third ∈ R.
L178109
An exact proof term for the current goal is one_third_in_R.
L178109
We prove the intermediate claim H23R: two_thirds ∈ R.
L178111
An exact proof term for the current goal is two_thirds_in_R.
L178111
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L178113
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L178113
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L178115
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L178115
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L178117
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L178117
We prove the intermediate claim Hr6sRx: apply_fun r6s x ∈ R.
L178119
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r6s x) Hr6sI1).
L178119
We prove the intermediate claim Hr6s_bounds: Rle (minus_SNo 1) (apply_fun r6s x) ∧ Rle (apply_fun r6s x) (minus_SNo one_third).
L178121
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r6s x) Hm1R Hm13R Hr6sI1).
L178122
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r6s x).
L178124
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r6s x)) (Rle (apply_fun r6s x) (minus_SNo one_third)) Hr6s_bounds).
L178125
We prove the intermediate claim HhiI1: Rle (apply_fun r6s x) (minus_SNo one_third).
L178127
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r6s x)) (Rle (apply_fun r6s x) (minus_SNo one_third)) Hr6s_bounds).
L178128
We prove the intermediate claim Hr7Rx: add_SNo (apply_fun r6s x) one_third ∈ R.
L178130
An exact proof term for the current goal is (real_add_SNo (apply_fun r6s x) Hr6sRx one_third H13R).
L178130
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r6s x) one_third).
L178132
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r6s x) one_third Hm1R Hr6sRx H13R Hm1le).
L178132
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r6s x) one_third).
L178134
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L178134
An exact proof term for the current goal is Hlow_tmp.
L178135
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r6s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L178137
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r6s x) (minus_SNo one_third) one_third Hr6sRx Hm13R H13R HhiI1).
L178137
We prove the intermediate claim H13S: SNo one_third.
L178139
An exact proof term for the current goal is (real_SNo one_third H13R).
L178139
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r6s x) one_third) 0.
L178141
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L178141
An exact proof term for the current goal is Hup0_tmp.
L178142
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L178144
An exact proof term for the current goal is Rle_0_two_thirds.
L178144
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r6s x) one_third) two_thirds.
L178146
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r6s x) one_third) 0 two_thirds Hup0 H0le23).
L178146
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r6s x) one_third) Hm23R H23R Hr7Rx Hlow Hup).
L178150
Assume HxnotB7: ¬ (x ∈ B7).
L178150
We prove the intermediate claim HC7_cases: x ∈ C7 ∨ ¬ (x ∈ C7).
L178152
An exact proof term for the current goal is (xm (x ∈ C7)).
L178152
Apply (HC7_cases (apply_fun r7 x ∈ I2)) to the current goal.
L178154
Assume HxC7: x ∈ C7.
L178154
We prove the intermediate claim Hu7eq: apply_fun u7 x = one_third.
L178156
An exact proof term for the current goal is (Hu7_on_C7 x HxC7).
L178156
We prove the intermediate claim Hr7eq: apply_fun r7 x = add_SNo (apply_fun r6s x) (minus_SNo one_third).
L178158
rewrite the current goal using (Hr7_apply x HxA) (from left to right).
L178158
rewrite the current goal using Hu7eq (from left to right) at position 1.
Use reflexivity.
L178160
rewrite the current goal using Hr7eq (from left to right).
L178161
We prove the intermediate claim Hr6sI3I: apply_fun r6s x ∈ I3 ∩ I.
L178163
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r6s x0 ∈ I3 ∩ I) x HxC7).
L178163
We prove the intermediate claim Hr6sI3: apply_fun r6s x ∈ I3.
L178165
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r6s x) Hr6sI3I).
L178165
We prove the intermediate claim H13R: one_third ∈ R.
L178167
An exact proof term for the current goal is one_third_in_R.
L178167
We prove the intermediate claim H23R: two_thirds ∈ R.
L178169
An exact proof term for the current goal is two_thirds_in_R.
L178169
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L178171
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L178171
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L178173
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L178173
We prove the intermediate claim Hr6sRx: apply_fun r6s x ∈ R.
L178175
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r6s x) Hr6sI3).
L178175
We prove the intermediate claim Hr6s_bounds: Rle one_third (apply_fun r6s x) ∧ Rle (apply_fun r6s x) 1.
L178177
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r6s x) H13R real_1 Hr6sI3).
L178177
We prove the intermediate claim HloI3: Rle one_third (apply_fun r6s x).
L178179
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r6s x)) (Rle (apply_fun r6s x) 1) Hr6s_bounds).
L178179
We prove the intermediate claim HhiI3: Rle (apply_fun r6s x) 1.
L178181
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r6s x)) (Rle (apply_fun r6s x) 1) Hr6s_bounds).
L178181
We prove the intermediate claim Hr7Rx: add_SNo (apply_fun r6s x) (minus_SNo one_third) ∈ R.
L178183
An exact proof term for the current goal is (real_add_SNo (apply_fun r6s x) Hr6sRx (minus_SNo one_third) Hm13R).
L178183
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r6s x) (minus_SNo one_third)).
L178186
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r6s x) (minus_SNo one_third) H13R Hr6sRx Hm13R HloI3).
L178187
We prove the intermediate claim H13S: SNo one_third.
L178189
An exact proof term for the current goal is (real_SNo one_third H13R).
L178189
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r6s x) (minus_SNo one_third)).
L178191
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L178191
An exact proof term for the current goal is H0le_tmp.
L178192
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L178194
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L178194
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r6s x) (minus_SNo one_third)).
L178196
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r6s x) (minus_SNo one_third)) Hm23le0 H0le).
L178198
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r6s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L178201
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r6s x) 1 (minus_SNo one_third) Hr6sRx real_1 Hm13R HhiI3).
L178202
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L178204
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L178204
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L178205
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r6s x) (minus_SNo one_third)) two_thirds.
L178207
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r6s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L178210
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r6s x) (minus_SNo one_third)) Hm23R H23R Hr7Rx Hlow Hup).
L178214
Assume HxnotC7: ¬ (x ∈ C7).
L178214
We prove the intermediate claim HxX: x ∈ X.
L178216
An exact proof term for the current goal is (HAsubX x HxA).
L178216
We prove the intermediate claim HnotI1: ¬ (apply_fun r6s x ∈ I1).
L178218
Assume Hr6sI1': apply_fun r6s x ∈ I1.
L178218
We prove the intermediate claim Hr6sI1I: apply_fun r6s x ∈ I1 ∩ I.
L178220
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r6s x) Hr6sI1' Hr6sIx).
L178220
We prove the intermediate claim HxB7': x ∈ B7.
L178222
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r6s x0 ∈ I1 ∩ I) x HxA Hr6sI1I).
L178222
Apply FalseE to the current goal.
L178223
An exact proof term for the current goal is (HxnotB7 HxB7').
L178224
We prove the intermediate claim HnotI3: ¬ (apply_fun r6s x ∈ I3).
L178226
Assume Hr6sI3': apply_fun r6s x ∈ I3.
L178226
We prove the intermediate claim Hr6sI3I: apply_fun r6s x ∈ I3 ∩ I.
L178228
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r6s x) Hr6sI3' Hr6sIx).
L178228
We prove the intermediate claim HxC7': x ∈ C7.
L178230
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r6s x0 ∈ I3 ∩ I) x HxA Hr6sI3I).
L178230
Apply FalseE to the current goal.
L178231
An exact proof term for the current goal is (HxnotC7 HxC7').
L178232
We prove the intermediate claim Hr6sRx: apply_fun r6s x ∈ R.
L178234
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r6s x) Hr6sIx).
L178234
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L178236
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L178236
We prove the intermediate claim H13R: one_third ∈ R.
L178238
An exact proof term for the current goal is one_third_in_R.
L178238
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L178240
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L178240
We prove the intermediate claim Hr6s_boundsI: Rle (minus_SNo 1) (apply_fun r6s x) ∧ Rle (apply_fun r6s x) 1.
L178242
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r6s x) Hm1R real_1 Hr6sIx).
L178242
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r6s x) (minus_SNo 1)).
L178244
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r6s x) (andEL (Rle (minus_SNo 1) (apply_fun r6s x)) (Rle (apply_fun r6s x) 1) Hr6s_boundsI)).
L178245
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r6s x)).
L178247
An exact proof term for the current goal is (RleE_nlt (apply_fun r6s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r6s x)) (Rle (apply_fun r6s x) 1) Hr6s_boundsI)).
L178248
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r6s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r6s x).
L178250
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r6s x) Hm1R Hm13R Hr6sRx HnotI1).
L178251
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r6s x).
L178253
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r6s x))) to the current goal.
L178254
Assume Hbad: Rlt (apply_fun r6s x) (minus_SNo 1).
L178254
Apply FalseE to the current goal.
L178255
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L178257
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r6s x).
L178257
An exact proof term for the current goal is Hok.
L178258
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r6s x) (minus_SNo one_third)).
L178260
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r6s x) Hm13lt_fx).
L178260
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r6s x) one_third ∨ Rlt 1 (apply_fun r6s x).
L178262
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r6s x) H13R real_1 Hr6sRx HnotI3).
L178263
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r6s x) one_third.
L178265
Apply (HnotI3_cases (Rlt (apply_fun r6s x) one_third)) to the current goal.
L178266
Assume Hok: Rlt (apply_fun r6s x) one_third.
L178266
An exact proof term for the current goal is Hok.
L178268
Assume Hbad: Rlt 1 (apply_fun r6s x).
L178268
Apply FalseE to the current goal.
L178269
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L178270
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r6s x)).
L178272
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r6s x) one_third Hfx_lt_13).
L178272
We prove the intermediate claim Hr6sI0: apply_fun r6s x ∈ I0.
L178274
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L178275
We prove the intermediate claim HxSep: apply_fun r6s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L178277
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r6s x) Hr6sRx (andI (¬ (Rlt (apply_fun r6s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r6s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L178281
rewrite the current goal using HI0_def (from left to right).
L178282
An exact proof term for the current goal is HxSep.
L178283
We prove the intermediate claim Hu7contI0: continuous_map X Tx I0 T0 u7.
L178285
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u7) (∀x0 : set, x0 ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x0 = minus_SNo one_third) Hu7AB).
L178289
We prove the intermediate claim Hu7funI0: function_on u7 X I0.
L178291
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u7 Hu7contI0).
L178291
We prove the intermediate claim Hu7xI0: apply_fun u7 x ∈ I0.
L178293
An exact proof term for the current goal is (Hu7funI0 x HxX).
L178293
We prove the intermediate claim Hu7xR: apply_fun u7 x ∈ R.
L178295
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u7 x) Hu7xI0).
L178295
We prove the intermediate claim Hm_u7x_R: minus_SNo (apply_fun u7 x) ∈ R.
L178297
An exact proof term for the current goal is (real_minus_SNo (apply_fun u7 x) Hu7xR).
L178297
rewrite the current goal using (Hr7_apply x HxA) (from left to right).
L178298
We prove the intermediate claim Hr7xR: add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x)) ∈ R.
L178300
An exact proof term for the current goal is (real_add_SNo (apply_fun r6s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r6s x) Hr6sI0) (minus_SNo (apply_fun u7 x)) Hm_u7x_R).
L178302
We prove the intermediate claim H23R: two_thirds ∈ R.
L178304
An exact proof term for the current goal is two_thirds_in_R.
L178304
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L178306
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L178306
We prove the intermediate claim Hr6s_bounds0: Rle (minus_SNo one_third) (apply_fun r6s x) ∧ Rle (apply_fun r6s x) one_third.
L178308
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r6s x) Hm13R H13R Hr6sI0).
L178308
We prove the intermediate claim Hu7_bounds0: Rle (minus_SNo one_third) (apply_fun u7 x) ∧ Rle (apply_fun u7 x) one_third.
L178310
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u7 x) Hm13R H13R Hu7xI0).
L178310
We prove the intermediate claim Hm13_le_r6s: Rle (minus_SNo one_third) (apply_fun r6s x).
L178312
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r6s x)) (Rle (apply_fun r6s x) one_third) Hr6s_bounds0).
L178312
We prove the intermediate claim Hr6s_le_13: Rle (apply_fun r6s x) one_third.
L178314
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r6s x)) (Rle (apply_fun r6s x) one_third) Hr6s_bounds0).
L178314
We prove the intermediate claim Hm13_le_u7x: Rle (minus_SNo one_third) (apply_fun u7 x).
L178316
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u7 x)) (Rle (apply_fun u7 x) one_third) Hu7_bounds0).
L178316
We prove the intermediate claim Hu7x_le_13: Rle (apply_fun u7 x) one_third.
L178318
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u7 x)) (Rle (apply_fun u7 x) one_third) Hu7_bounds0).
L178318
We prove the intermediate claim Hm13_le_mu7: Rle (minus_SNo one_third) (minus_SNo (apply_fun u7 x)).
L178320
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u7 x) one_third Hu7x_le_13).
L178320
We prove the intermediate claim Hmu7_le_13: Rle (minus_SNo (apply_fun u7 x)) one_third.
L178322
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u7 x)) (minus_SNo (minus_SNo one_third)).
L178323
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u7 x) Hm13_le_u7x).
L178323
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L178324
An exact proof term for the current goal is Htmp.
L178325
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u7 x))).
L178328
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u7 x)) Hm13R Hm13R Hm_u7x_R Hm13_le_mu7).
L178329
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u7 x))) (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))).
L178332
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r6s x) (minus_SNo (apply_fun u7 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r6s x) Hr6sI0) Hm_u7x_R Hm13_le_r6s).
L178336
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))).
L178339
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u7 x))) (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))) Hlow1 Hlow2).
L178342
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))).
L178344
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L178344
An exact proof term for the current goal is Hlow_tmp.
L178345
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))) (add_SNo (apply_fun r6s x) one_third).
L178348
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r6s x) (minus_SNo (apply_fun u7 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r6s x) Hr6sI0) Hm_u7x_R H13R Hmu7_le_13).
L178350
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r6s x) one_third) (add_SNo one_third one_third).
L178353
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r6s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r6s x) Hr6sI0) H13R H13R Hr6s_le_13).
L178355
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))) (add_SNo one_third one_third).
L178358
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))) (add_SNo (apply_fun r6s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L178361
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L178363
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))) two_thirds.
L178365
rewrite the current goal using Hdef23 (from left to right) at position 1.
L178365
An exact proof term for the current goal is Hup_tmp.
L178366
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))) Hm23R H23R Hr7xR Hlow Hup).
L178369
Set r7s to be the term compose_fun A r7 (div_const_fun den).
L178370
We prove the intermediate claim Hr7s_cont: continuous_map A Ta I Ti r7s.
L178372
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
L178373
An exact proof term for the current goal is R_standard_topology_is_topology.
L178373
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L178375
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L178375
We prove the intermediate claim Hr7s_contR: continuous_map A Ta R R_standard_topology r7s.
L178377
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r7 (div_const_fun den) Hr7_cont Hdivcont).
L178378
We prove the intermediate claim Hr7s_I: ∀x : set, x ∈ A → apply_fun r7s x ∈ I.
L178380
Let x be given.
L178380
Assume HxA: x ∈ A.
L178380
We prove the intermediate claim Hr7xI2: apply_fun r7 x ∈ I2.
L178382
An exact proof term for the current goal is (Hr7_range x HxA).
L178382
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L178384
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L178384
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L178386
An exact proof term for the current goal is (real_minus_SNo den H23R).
L178386
We prove the intermediate claim Hr7xR: apply_fun r7 x ∈ R.
L178388
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r7 x) Hr7xI2).
L178388
We prove the intermediate claim Hr7xS: SNo (apply_fun r7 x).
L178390
An exact proof term for the current goal is (real_SNo (apply_fun r7 x) Hr7xR).
L178390
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r7 x) ∧ Rle (apply_fun r7 x) den.
L178392
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r7 x) HmdenR H23R Hr7xI2).
L178392
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r7 x).
L178394
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r7 x)) (Rle (apply_fun r7 x) den) Hbounds).
L178396
We prove the intermediate claim Hhi: Rle (apply_fun r7 x) den.
L178398
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r7 x)) (Rle (apply_fun r7 x) den) Hbounds).
L178400
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r7 x)).
L178402
An exact proof term for the current goal is (RleE_nlt (apply_fun r7 x) den Hhi).
L178402
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r7 x) (minus_SNo den)).
L178404
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r7 x) Hlo).
L178404
We prove the intermediate claim HyEq: apply_fun r7s x = div_SNo (apply_fun r7 x) den.
L178406
rewrite the current goal using (compose_fun_apply A r7 (div_const_fun den) x HxA) (from left to right).
L178406
rewrite the current goal using (div_const_fun_apply den (apply_fun r7 x) H23R Hr7xR) (from left to right).
Use reflexivity.
L178408
We prove the intermediate claim HyR: apply_fun r7s x ∈ R.
L178410
rewrite the current goal using HyEq (from left to right).
L178410
An exact proof term for the current goal is (real_div_SNo (apply_fun r7 x) Hr7xR den H23R).
L178411
We prove the intermediate claim HyS: SNo (apply_fun r7s x).
L178413
An exact proof term for the current goal is (real_SNo (apply_fun r7s x) HyR).
L178413
We prove the intermediate claim Hy_le_1: Rle (apply_fun r7s x) 1.
L178415
Apply (RleI (apply_fun r7s x) 1 HyR real_1) to the current goal.
L178415
We will prove ¬ (Rlt 1 (apply_fun r7s x)).
L178416
Assume H1lt: Rlt 1 (apply_fun r7s x).
L178417
We prove the intermediate claim H1lty: 1 < apply_fun r7s x.
L178419
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r7s x) H1lt).
L178419
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r7s x) den.
L178421
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r7s x) den SNo_1 HyS H23S H23pos H1lty).
L178421
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r7s x) den.
L178423
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L178423
An exact proof term for the current goal is HmulLt.
L178424
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r7s x) den = apply_fun r7 x.
L178426
rewrite the current goal using HyEq (from left to right) at position 1.
L178426
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r7 x) den Hr7xS H23S H23ne0).
L178427
We prove the intermediate claim Hden_lt_r7x: den < apply_fun r7 x.
L178429
rewrite the current goal using HmulEq (from right to left).
L178429
An exact proof term for the current goal is HmulLt'.
L178430
We prove the intermediate claim Hbad: Rlt den (apply_fun r7 x).
L178432
An exact proof term for the current goal is (RltI den (apply_fun r7 x) H23R Hr7xR Hden_lt_r7x).
L178432
An exact proof term for the current goal is (Hnlt_hi Hbad).
L178433
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r7s x).
L178435
Apply (RleI (minus_SNo 1) (apply_fun r7s x) Hm1R HyR) to the current goal.
L178435
We will prove ¬ (Rlt (apply_fun r7s x) (minus_SNo 1)).
L178436
Assume Hylt: Rlt (apply_fun r7s x) (minus_SNo 1).
L178437
We prove the intermediate claim Hylts: apply_fun r7s x < minus_SNo 1.
L178439
An exact proof term for the current goal is (RltE_lt (apply_fun r7s x) (minus_SNo 1) Hylt).
L178439
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r7s x) den < mul_SNo (minus_SNo 1) den.
L178441
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r7s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L178442
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r7s x) den = apply_fun r7 x.
L178444
rewrite the current goal using HyEq (from left to right) at position 1.
L178444
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r7 x) den Hr7xS H23S H23ne0).
L178445
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L178447
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L178447
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L178449
We prove the intermediate claim Hr7x_lt_mden: apply_fun r7 x < minus_SNo den.
L178451
rewrite the current goal using HmulEq (from right to left).
L178451
rewrite the current goal using HrhsEq (from right to left).
L178452
An exact proof term for the current goal is HmulLt.
L178453
We prove the intermediate claim Hbad: Rlt (apply_fun r7 x) (minus_SNo den).
L178455
An exact proof term for the current goal is (RltI (apply_fun r7 x) (minus_SNo den) Hr7xR HmdenR Hr7x_lt_mden).
L178455
An exact proof term for the current goal is (Hnlt_lo Hbad).
L178456
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r7s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L178458
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r7s I Hr7s_contR HIcR Hr7s_I).
L178459
We prove the intermediate claim Hex_u8: ∃u8 : set, continuous_map X Tx I0 T0 u8 ∧ (∀x : set, x ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r7s ((closed_interval one_third 1) ∩ I) → apply_fun u8 x = one_third).
(*** thirteenth correction step scaffold: build u8 and g8 from residual r7s ***)
L178469
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r7s Hnorm HA Hr7s_cont).
L178469
Apply Hex_u8 to the current goal.
L178470
Let u8 be given.
L178471
Assume Hu8.
L178471
We prove the intermediate claim Hu8contI0: continuous_map X Tx I0 T0 u8.
L178473
We prove the intermediate claim Hu8AB: continuous_map X Tx I0 T0 u8 ∧ (∀x : set, x ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x = minus_SNo one_third).
L178477
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u8 ∧ (∀x : set, x ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r7s ((closed_interval one_third 1) ∩ I) → apply_fun u8 x = one_third) Hu8).
L178483
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u8) (∀x : set, x ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x = minus_SNo one_third) Hu8AB).
L178488
We prove the intermediate claim Hu8contR: continuous_map X Tx R R_standard_topology u8.
L178490
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L178491
We prove the intermediate claim HI0subR: I0 ⊆ R.
L178493
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L178493
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u8 Hu8contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L178501
Set den8 to be the term mul_SNo den7 den.
L178502
We prove the intermediate claim Hden8R: den8 ∈ R.
L178504
An exact proof term for the current goal is (real_mul_SNo den7 Hden7R den H23R).
L178504
We prove the intermediate claim Hden8pos: 0 < den8.
L178506
We prove the intermediate claim Hden7S: SNo den7.
L178507
An exact proof term for the current goal is (real_SNo den7 Hden7R).
L178507
An exact proof term for the current goal is (mul_SNo_pos_pos den7 den Hden7S H23S Hden7pos HdenPos).
L178508
Set u8s to be the term compose_fun X u8 (mul_const_fun den8).
L178509
We prove the intermediate claim Hu8s_cont: continuous_map X Tx R R_standard_topology u8s.
L178511
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u8 den8 HTx Hu8contR Hden8R Hden8pos).
L178511
Set g8 to be the term compose_fun X (pair_map X g7 u8s) add_fun_R.
L178512
We prove the intermediate claim Hg8cont: continuous_map X Tx R R_standard_topology g8.
L178514
An exact proof term for the current goal is (add_two_continuous_R X Tx g7 u8s HTx Hg7cont Hu8s_cont).
L178514
We prove the intermediate claim Hu8contA: continuous_map A Ta R R_standard_topology u8.
(*** fourteenth correction step scaffold: residual r8 on A and scaling r8s ***)
L178518
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u8 A HTx HAsubX Hu8contR).
L178518
Set u8neg to be the term compose_fun A u8 neg_fun.
L178519
We prove the intermediate claim Hu8neg_cont: continuous_map A Ta R R_standard_topology u8neg.
L178521
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u8 neg_fun Hu8contA Hnegcont).
L178522
We prove the intermediate claim Hr7s_contR: continuous_map A Ta R R_standard_topology r7s.
L178524
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r7s Hr7s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L178525
Set r8 to be the term compose_fun A (pair_map A r7s u8neg) add_fun_R.
L178526
We prove the intermediate claim Hr8_cont: continuous_map A Ta R R_standard_topology r8.
L178528
An exact proof term for the current goal is (add_two_continuous_R A Ta r7s u8neg HTa Hr7s_contR Hu8neg_cont).
L178528
We prove the intermediate claim Hr8_apply: ∀x : set, x ∈ A → apply_fun r8 x = add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x)).
L178531
Let x be given.
L178531
Assume HxA: x ∈ A.
L178531
We prove the intermediate claim Hpimg: apply_fun (pair_map A r7s u8neg) x ∈ setprod R R.
L178533
rewrite the current goal using (pair_map_apply A R R r7s u8neg x HxA) (from left to right).
L178533
We prove the intermediate claim Hr7sxI: apply_fun r7s x ∈ I.
L178535
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r7s Hr7s_cont x HxA).
L178535
We prove the intermediate claim Hr7sxR: apply_fun r7s x ∈ R.
L178537
An exact proof term for the current goal is (HIcR (apply_fun r7s x) Hr7sxI).
L178537
We prove the intermediate claim Hu8negRx: apply_fun u8neg x ∈ R.
L178539
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u8neg Hu8neg_cont x HxA).
L178539
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r7s x) (apply_fun u8neg x) Hr7sxR Hu8negRx).
L178540
rewrite the current goal using (compose_fun_apply A (pair_map A r7s u8neg) add_fun_R x HxA) (from left to right).
L178541
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r7s u8neg) x) Hpimg) (from left to right) at position 1.
L178542
rewrite the current goal using (pair_map_apply A R R r7s u8neg x HxA) (from left to right).
L178543
rewrite the current goal using (tuple_2_0_eq (apply_fun r7s x) (apply_fun u8neg x)) (from left to right).
L178544
rewrite the current goal using (tuple_2_1_eq (apply_fun r7s x) (apply_fun u8neg x)) (from left to right).
L178545
rewrite the current goal using (compose_fun_apply A u8 neg_fun x HxA) (from left to right) at position 1.
L178546
We prove the intermediate claim Hu8Rx: apply_fun u8 x ∈ R.
L178548
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u8 Hu8contA x HxA).
L178548
rewrite the current goal using (neg_fun_apply (apply_fun u8 x) Hu8Rx) (from left to right) at position 1.
Use reflexivity.
L178550
We prove the intermediate claim Hr8_range: ∀x : set, x ∈ A → apply_fun r8 x ∈ I2.
L178552
We prove the intermediate claim Hu8AB: continuous_map X Tx I0 T0 u8 ∧ (∀x0 : set, x0 ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x0 = minus_SNo one_third).
L178556
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u8 ∧ (∀x0 : set, x0 ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r7s ((closed_interval one_third 1) ∩ I) → apply_fun u8 x0 = one_third) Hu8).
L178562
We prove the intermediate claim Hu8_on_B8: ∀x0 : set, x0 ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x0 = minus_SNo one_third.
L178566
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u8) (∀x0 : set, x0 ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x0 = minus_SNo one_third) Hu8AB).
L178570
We prove the intermediate claim Hu8_on_C8: ∀x0 : set, x0 ∈ preimage_of A r7s ((closed_interval one_third 1) ∩ I) → apply_fun u8 x0 = one_third.
L178574
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u8 ∧ (∀x0 : set, x0 ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r7s ((closed_interval one_third 1) ∩ I) → apply_fun u8 x0 = one_third) Hu8).
L178580
Let x be given.
L178581
Assume HxA: x ∈ A.
L178581
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L178582
Set I3 to be the term closed_interval one_third 1.
L178583
Set B8 to be the term preimage_of A r7s (I1 ∩ I).
L178584
Set C8 to be the term preimage_of A r7s (I3 ∩ I).
L178585
We prove the intermediate claim Hr7sIx: apply_fun r7s x ∈ I.
L178587
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r7s Hr7s_cont x HxA).
L178587
We prove the intermediate claim HB8_cases: x ∈ B8 ∨ ¬ (x ∈ B8).
L178589
An exact proof term for the current goal is (xm (x ∈ B8)).
L178589
Apply (HB8_cases (apply_fun r8 x ∈ I2)) to the current goal.
L178591
Assume HxB8: x ∈ B8.
L178591
We prove the intermediate claim Hu8eq: apply_fun u8 x = minus_SNo one_third.
L178593
An exact proof term for the current goal is (Hu8_on_B8 x HxB8).
L178593
We prove the intermediate claim Hr8eq: apply_fun r8 x = add_SNo (apply_fun r7s x) one_third.
L178595
rewrite the current goal using (Hr8_apply x HxA) (from left to right).
L178595
rewrite the current goal using Hu8eq (from left to right) at position 1.
L178596
We prove the intermediate claim H13R: one_third ∈ R.
L178598
An exact proof term for the current goal is one_third_in_R.
L178598
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L178600
rewrite the current goal using Hr8eq (from left to right).
L178601
We prove the intermediate claim Hr7sI1I: apply_fun r7s x ∈ I1 ∩ I.
L178603
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r7s x0 ∈ I1 ∩ I) x HxB8).
L178603
We prove the intermediate claim Hr7sI1: apply_fun r7s x ∈ I1.
L178605
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r7s x) Hr7sI1I).
L178605
We prove the intermediate claim H13R: one_third ∈ R.
L178607
An exact proof term for the current goal is one_third_in_R.
L178607
We prove the intermediate claim H23R: two_thirds ∈ R.
L178609
An exact proof term for the current goal is two_thirds_in_R.
L178609
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L178611
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L178611
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L178613
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L178613
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L178615
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L178615
We prove the intermediate claim Hr7sRx: apply_fun r7s x ∈ R.
L178617
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r7s x) Hr7sI1).
L178617
We prove the intermediate claim Hr7s_bounds: Rle (minus_SNo 1) (apply_fun r7s x) ∧ Rle (apply_fun r7s x) (minus_SNo one_third).
L178619
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r7s x) Hm1R Hm13R Hr7sI1).
L178620
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r7s x).
L178622
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r7s x)) (Rle (apply_fun r7s x) (minus_SNo one_third)) Hr7s_bounds).
L178623
We prove the intermediate claim HhiI1: Rle (apply_fun r7s x) (minus_SNo one_third).
L178625
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r7s x)) (Rle (apply_fun r7s x) (minus_SNo one_third)) Hr7s_bounds).
L178626
We prove the intermediate claim Hr8xR: add_SNo (apply_fun r7s x) one_third ∈ R.
L178628
An exact proof term for the current goal is (real_add_SNo (apply_fun r7s x) Hr7sRx one_third H13R).
L178628
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r7s x) one_third).
L178630
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r7s x) one_third Hm1R Hr7sRx H13R Hm1le).
L178630
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r7s x) one_third).
L178632
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L178632
An exact proof term for the current goal is Hlow_tmp.
L178633
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r7s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L178635
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r7s x) (minus_SNo one_third) one_third Hr7sRx Hm13R H13R HhiI1).
L178635
We prove the intermediate claim H13S: SNo one_third.
L178637
An exact proof term for the current goal is (real_SNo one_third H13R).
L178637
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r7s x) one_third) 0.
L178639
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L178639
An exact proof term for the current goal is Hup0_tmp.
L178640
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L178642
An exact proof term for the current goal is Rle_0_two_thirds.
L178642
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r7s x) one_third) two_thirds.
L178644
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r7s x) one_third) 0 two_thirds Hup0 H0le23).
L178644
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r7s x) one_third) Hm23R H23R Hr8xR Hlow Hup).
L178648
Assume HxnotB8: ¬ (x ∈ B8).
L178648
We prove the intermediate claim HC8_cases: x ∈ C8 ∨ ¬ (x ∈ C8).
L178650
An exact proof term for the current goal is (xm (x ∈ C8)).
L178650
Apply (HC8_cases (apply_fun r8 x ∈ I2)) to the current goal.
L178652
Assume HxC8: x ∈ C8.
L178652
We prove the intermediate claim Hu8eq: apply_fun u8 x = one_third.
L178654
An exact proof term for the current goal is (Hu8_on_C8 x HxC8).
L178654
We prove the intermediate claim Hr8eq: apply_fun r8 x = add_SNo (apply_fun r7s x) (minus_SNo one_third).
L178656
rewrite the current goal using (Hr8_apply x HxA) (from left to right).
L178656
rewrite the current goal using Hu8eq (from left to right) at position 1.
Use reflexivity.
L178658
rewrite the current goal using Hr8eq (from left to right).
L178659
We prove the intermediate claim Hr7sI3I: apply_fun r7s x ∈ I3 ∩ I.
L178661
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r7s x0 ∈ I3 ∩ I) x HxC8).
L178661
We prove the intermediate claim Hr7sI3: apply_fun r7s x ∈ I3.
L178663
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r7s x) Hr7sI3I).
L178663
We prove the intermediate claim H13R: one_third ∈ R.
L178665
An exact proof term for the current goal is one_third_in_R.
L178665
We prove the intermediate claim H23R: two_thirds ∈ R.
L178667
An exact proof term for the current goal is two_thirds_in_R.
L178667
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L178669
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L178669
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L178671
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L178671
We prove the intermediate claim Hr7sRx: apply_fun r7s x ∈ R.
L178673
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r7s x) Hr7sI3).
L178673
We prove the intermediate claim Hr7s_bounds: Rle one_third (apply_fun r7s x) ∧ Rle (apply_fun r7s x) 1.
L178675
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r7s x) H13R real_1 Hr7sI3).
L178675
We prove the intermediate claim HloI3: Rle one_third (apply_fun r7s x).
L178677
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r7s x)) (Rle (apply_fun r7s x) 1) Hr7s_bounds).
L178677
We prove the intermediate claim HhiI3: Rle (apply_fun r7s x) 1.
L178679
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r7s x)) (Rle (apply_fun r7s x) 1) Hr7s_bounds).
L178679
We prove the intermediate claim Hr8xR: add_SNo (apply_fun r7s x) (minus_SNo one_third) ∈ R.
L178681
An exact proof term for the current goal is (real_add_SNo (apply_fun r7s x) Hr7sRx (minus_SNo one_third) Hm13R).
L178681
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r7s x) (minus_SNo one_third)).
L178684
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r7s x) (minus_SNo one_third) H13R Hr7sRx Hm13R HloI3).
L178685
We prove the intermediate claim H13S: SNo one_third.
L178687
An exact proof term for the current goal is (real_SNo one_third H13R).
L178687
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r7s x) (minus_SNo one_third)).
L178689
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L178689
An exact proof term for the current goal is H0le_tmp.
L178690
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L178692
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L178692
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r7s x) (minus_SNo one_third)).
L178694
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r7s x) (minus_SNo one_third)) Hm23le0 H0le).
L178696
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r7s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L178699
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r7s x) 1 (minus_SNo one_third) Hr7sRx real_1 Hm13R HhiI3).
L178700
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L178702
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L178702
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L178703
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r7s x) (minus_SNo one_third)) two_thirds.
L178705
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r7s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L178708
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r7s x) (minus_SNo one_third)) Hm23R H23R Hr8xR Hlow Hup).
L178712
Assume HxnotC8: ¬ (x ∈ C8).
L178712
We prove the intermediate claim HxX: x ∈ X.
L178714
An exact proof term for the current goal is (HAsubX x HxA).
L178714
We prove the intermediate claim HnotI1: ¬ (apply_fun r7s x ∈ I1).
L178716
Assume Hr7sI1': apply_fun r7s x ∈ I1.
L178716
We prove the intermediate claim Hr7sI1I: apply_fun r7s x ∈ I1 ∩ I.
L178718
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r7s x) Hr7sI1' Hr7sIx).
L178718
We prove the intermediate claim HxB8': x ∈ B8.
L178720
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r7s x0 ∈ I1 ∩ I) x HxA Hr7sI1I).
L178720
Apply FalseE to the current goal.
L178721
An exact proof term for the current goal is (HxnotB8 HxB8').
L178722
We prove the intermediate claim HnotI3: ¬ (apply_fun r7s x ∈ I3).
L178724
Assume Hr7sI3': apply_fun r7s x ∈ I3.
L178724
We prove the intermediate claim Hr7sI3I: apply_fun r7s x ∈ I3 ∩ I.
L178726
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r7s x) Hr7sI3' Hr7sIx).
L178726
We prove the intermediate claim HxC8': x ∈ C8.
L178728
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r7s x0 ∈ I3 ∩ I) x HxA Hr7sI3I).
L178728
Apply FalseE to the current goal.
L178729
An exact proof term for the current goal is (HxnotC8 HxC8').
L178730
We prove the intermediate claim Hr7sRx: apply_fun r7s x ∈ R.
L178732
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r7s x) Hr7sIx).
L178732
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L178734
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L178734
We prove the intermediate claim H13R: one_third ∈ R.
L178736
An exact proof term for the current goal is one_third_in_R.
L178736
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L178738
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L178738
We prove the intermediate claim Hr7s_boundsI: Rle (minus_SNo 1) (apply_fun r7s x) ∧ Rle (apply_fun r7s x) 1.
L178740
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r7s x) Hm1R real_1 Hr7sIx).
L178740
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r7s x) (minus_SNo 1)).
L178742
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r7s x) (andEL (Rle (minus_SNo 1) (apply_fun r7s x)) (Rle (apply_fun r7s x) 1) Hr7s_boundsI)).
L178743
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r7s x)).
L178745
An exact proof term for the current goal is (RleE_nlt (apply_fun r7s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r7s x)) (Rle (apply_fun r7s x) 1) Hr7s_boundsI)).
L178746
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r7s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r7s x).
L178748
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r7s x) Hm1R Hm13R Hr7sRx HnotI1).
L178749
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r7s x).
L178751
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r7s x))) to the current goal.
L178752
Assume Hbad: Rlt (apply_fun r7s x) (minus_SNo 1).
L178752
Apply FalseE to the current goal.
L178753
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L178755
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r7s x).
L178755
An exact proof term for the current goal is Hok.
L178756
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r7s x) (minus_SNo one_third)).
L178758
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r7s x) Hm13lt_fx).
L178758
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r7s x) one_third ∨ Rlt 1 (apply_fun r7s x).
L178760
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r7s x) H13R real_1 Hr7sRx HnotI3).
L178761
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r7s x) one_third.
L178763
Apply (HnotI3_cases (Rlt (apply_fun r7s x) one_third)) to the current goal.
L178764
Assume Hok: Rlt (apply_fun r7s x) one_third.
L178764
An exact proof term for the current goal is Hok.
L178766
Assume Hbad: Rlt 1 (apply_fun r7s x).
L178766
Apply FalseE to the current goal.
L178767
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L178768
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r7s x)).
L178770
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r7s x) one_third Hfx_lt_13).
L178770
We prove the intermediate claim Hr7sI0: apply_fun r7s x ∈ I0.
L178772
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L178773
We prove the intermediate claim HxSep: apply_fun r7s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L178775
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r7s x) Hr7sRx (andI (¬ (Rlt (apply_fun r7s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r7s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L178779
rewrite the current goal using HI0_def (from left to right).
L178780
An exact proof term for the current goal is HxSep.
L178781
We prove the intermediate claim Hu8funI0: function_on u8 X I0.
L178783
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u8 Hu8contI0).
L178783
We prove the intermediate claim Hu8xI0: apply_fun u8 x ∈ I0.
L178785
An exact proof term for the current goal is (Hu8funI0 x HxX).
L178785
We prove the intermediate claim Hu8xR: apply_fun u8 x ∈ R.
L178787
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u8 x) Hu8xI0).
L178787
We prove the intermediate claim Hm_u8x_R: minus_SNo (apply_fun u8 x) ∈ R.
L178789
An exact proof term for the current goal is (real_minus_SNo (apply_fun u8 x) Hu8xR).
L178789
rewrite the current goal using (Hr8_apply x HxA) (from left to right).
L178790
We prove the intermediate claim Hr8xR: add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x)) ∈ R.
L178792
An exact proof term for the current goal is (real_add_SNo (apply_fun r7s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r7s x) Hr7sI0) (minus_SNo (apply_fun u8 x)) Hm_u8x_R).
L178794
We prove the intermediate claim H23R: two_thirds ∈ R.
L178796
An exact proof term for the current goal is two_thirds_in_R.
L178796
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L178798
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L178798
We prove the intermediate claim Hr7s_bounds0: Rle (minus_SNo one_third) (apply_fun r7s x) ∧ Rle (apply_fun r7s x) one_third.
L178800
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r7s x) Hm13R H13R Hr7sI0).
L178800
We prove the intermediate claim Hu8_bounds0: Rle (minus_SNo one_third) (apply_fun u8 x) ∧ Rle (apply_fun u8 x) one_third.
L178802
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u8 x) Hm13R H13R Hu8xI0).
L178802
We prove the intermediate claim Hm13_le_r7s: Rle (minus_SNo one_third) (apply_fun r7s x).
L178804
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r7s x)) (Rle (apply_fun r7s x) one_third) Hr7s_bounds0).
L178804
We prove the intermediate claim Hr7s_le_13: Rle (apply_fun r7s x) one_third.
L178806
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r7s x)) (Rle (apply_fun r7s x) one_third) Hr7s_bounds0).
L178806
We prove the intermediate claim Hm13_le_u8x: Rle (minus_SNo one_third) (apply_fun u8 x).
L178808
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u8 x)) (Rle (apply_fun u8 x) one_third) Hu8_bounds0).
L178808
We prove the intermediate claim Hu8x_le_13: Rle (apply_fun u8 x) one_third.
L178810
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u8 x)) (Rle (apply_fun u8 x) one_third) Hu8_bounds0).
L178810
We prove the intermediate claim Hm13_le_mu8: Rle (minus_SNo one_third) (minus_SNo (apply_fun u8 x)).
L178812
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u8 x) one_third Hu8x_le_13).
L178812
We prove the intermediate claim Hmu8_le_13: Rle (minus_SNo (apply_fun u8 x)) one_third.
L178814
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u8 x)) (minus_SNo (minus_SNo one_third)).
L178815
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u8 x) Hm13_le_u8x).
L178815
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L178816
An exact proof term for the current goal is Htmp.
L178817
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u8 x))).
L178820
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u8 x)) Hm13R Hm13R Hm_u8x_R Hm13_le_mu8).
L178821
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u8 x))) (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))).
L178824
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r7s x) (minus_SNo (apply_fun u8 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r7s x) Hr7sI0) Hm_u8x_R Hm13_le_r7s).
L178828
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))).
L178831
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u8 x))) (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))) Hlow1 Hlow2).
L178834
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))).
L178836
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L178836
An exact proof term for the current goal is Hlow_tmp.
L178837
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))) (add_SNo (apply_fun r7s x) one_third).
L178840
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r7s x) (minus_SNo (apply_fun u8 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r7s x) Hr7sI0) Hm_u8x_R H13R Hmu8_le_13).
L178842
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r7s x) one_third) (add_SNo one_third one_third).
L178845
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r7s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r7s x) Hr7sI0) H13R H13R Hr7s_le_13).
L178847
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))) (add_SNo one_third one_third).
L178850
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))) (add_SNo (apply_fun r7s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L178853
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L178855
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))) two_thirds.
L178857
rewrite the current goal using Hdef23 (from left to right) at position 1.
L178857
An exact proof term for the current goal is Hup_tmp.
L178858
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))) Hm23R H23R Hr8xR Hlow Hup).
L178861
Set r8s to be the term compose_fun A r8 (div_const_fun den).
L178862
We prove the intermediate claim Hr8s_cont: continuous_map A Ta I Ti r8s.
L178864
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
L178865
An exact proof term for the current goal is R_standard_topology_is_topology.
L178865
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L178867
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L178867
We prove the intermediate claim Hr8s_contR: continuous_map A Ta R R_standard_topology r8s.
L178869
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r8 (div_const_fun den) Hr8_cont Hdivcont).
L178870
We prove the intermediate claim Hr8s_I: ∀x : set, x ∈ A → apply_fun r8s x ∈ I.
L178872
Let x be given.
L178872
Assume HxA: x ∈ A.
L178872
We prove the intermediate claim Hr8xI2: apply_fun r8 x ∈ I2.
L178874
An exact proof term for the current goal is (Hr8_range x HxA).
L178874
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L178876
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L178876
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L178878
An exact proof term for the current goal is (real_minus_SNo den H23R).
L178878
We prove the intermediate claim Hr8xR: apply_fun r8 x ∈ R.
L178880
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r8 x) Hr8xI2).
L178880
We prove the intermediate claim Hr8xS: SNo (apply_fun r8 x).
L178882
An exact proof term for the current goal is (real_SNo (apply_fun r8 x) Hr8xR).
L178882
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r8 x) ∧ Rle (apply_fun r8 x) den.
L178884
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r8 x) HmdenR H23R Hr8xI2).
L178884
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r8 x).
L178886
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r8 x)) (Rle (apply_fun r8 x) den) Hbounds).
L178888
We prove the intermediate claim Hhi: Rle (apply_fun r8 x) den.
L178890
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r8 x)) (Rle (apply_fun r8 x) den) Hbounds).
L178892
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r8 x)).
L178894
An exact proof term for the current goal is (RleE_nlt (apply_fun r8 x) den Hhi).
L178894
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r8 x) (minus_SNo den)).
L178896
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r8 x) Hlo).
L178896
We prove the intermediate claim HyEq: apply_fun r8s x = div_SNo (apply_fun r8 x) den.
L178898
rewrite the current goal using (compose_fun_apply A r8 (div_const_fun den) x HxA) (from left to right).
L178898
rewrite the current goal using (div_const_fun_apply den (apply_fun r8 x) H23R Hr8xR) (from left to right).
Use reflexivity.
L178900
We prove the intermediate claim HyR: apply_fun r8s x ∈ R.
L178902
rewrite the current goal using HyEq (from left to right).
L178902
An exact proof term for the current goal is (real_div_SNo (apply_fun r8 x) Hr8xR den H23R).
L178903
We prove the intermediate claim HyS: SNo (apply_fun r8s x).
L178905
An exact proof term for the current goal is (real_SNo (apply_fun r8s x) HyR).
L178905
We prove the intermediate claim Hy_le_1: Rle (apply_fun r8s x) 1.
L178907
Apply (RleI (apply_fun r8s x) 1 HyR real_1) to the current goal.
L178907
We will prove ¬ (Rlt 1 (apply_fun r8s x)).
L178908
Assume H1lt: Rlt 1 (apply_fun r8s x).
L178909
We prove the intermediate claim H1lty: 1 < apply_fun r8s x.
L178911
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r8s x) H1lt).
L178911
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r8s x) den.
L178913
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r8s x) den SNo_1 HyS H23S H23pos H1lty).
L178913
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r8s x) den.
L178915
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L178915
An exact proof term for the current goal is HmulLt.
L178916
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r8s x) den = apply_fun r8 x.
L178918
rewrite the current goal using HyEq (from left to right) at position 1.
L178918
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r8 x) den Hr8xS H23S H23ne0).
L178919
We prove the intermediate claim Hden_lt_r8x: den < apply_fun r8 x.
L178921
rewrite the current goal using HmulEq (from right to left).
L178921
An exact proof term for the current goal is HmulLt'.
L178922
We prove the intermediate claim Hbad: Rlt den (apply_fun r8 x).
L178924
An exact proof term for the current goal is (RltI den (apply_fun r8 x) H23R Hr8xR Hden_lt_r8x).
L178924
An exact proof term for the current goal is (Hnlt_hi Hbad).
L178925
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r8s x).
L178927
Apply (RleI (minus_SNo 1) (apply_fun r8s x) Hm1R HyR) to the current goal.
L178927
We will prove ¬ (Rlt (apply_fun r8s x) (minus_SNo 1)).
L178928
Assume Hylt: Rlt (apply_fun r8s x) (minus_SNo 1).
L178929
We prove the intermediate claim Hylts: apply_fun r8s x < minus_SNo 1.
L178931
An exact proof term for the current goal is (RltE_lt (apply_fun r8s x) (minus_SNo 1) Hylt).
L178931
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r8s x) den < mul_SNo (minus_SNo 1) den.
L178933
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r8s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L178934
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r8s x) den = apply_fun r8 x.
L178936
rewrite the current goal using HyEq (from left to right) at position 1.
L178936
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r8 x) den Hr8xS H23S H23ne0).
L178937
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L178939
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L178939
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L178941
We prove the intermediate claim Hr8x_lt_mden: apply_fun r8 x < minus_SNo den.
L178943
rewrite the current goal using HmulEq (from right to left).
L178943
rewrite the current goal using HrhsEq (from right to left).
L178944
An exact proof term for the current goal is HmulLt.
L178945
We prove the intermediate claim Hbad: Rlt (apply_fun r8 x) (minus_SNo den).
L178947
An exact proof term for the current goal is (RltI (apply_fun r8 x) (minus_SNo den) Hr8xR HmdenR Hr8x_lt_mden).
L178947
An exact proof term for the current goal is (Hnlt_lo Hbad).
L178948
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r8s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L178950
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r8s I Hr8s_contR HIcR Hr8s_I).
L178951
We prove the intermediate claim Hex_u9: ∃u9 : set, continuous_map X Tx I0 T0 u9 ∧ (∀x : set, x ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r8s ((closed_interval one_third 1) ∩ I) → apply_fun u9 x = one_third).
(*** fifteenth correction step scaffold: build u9 and g9 from residual r8s ***)
L178961
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r8s Hnorm HA Hr8s_cont).
L178961
Apply Hex_u9 to the current goal.
L178962
Let u9 be given.
L178963
Assume Hu9.
L178963
We prove the intermediate claim Hu9contI0: continuous_map X Tx I0 T0 u9.
L178965
We prove the intermediate claim Hu9AB: continuous_map X Tx I0 T0 u9 ∧ (∀x : set, x ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x = minus_SNo one_third).
L178969
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u9 ∧ (∀x : set, x ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r8s ((closed_interval one_third 1) ∩ I) → apply_fun u9 x = one_third) Hu9).
L178975
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u9) (∀x : set, x ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x = minus_SNo one_third) Hu9AB).
L178980
We prove the intermediate claim Hu9contR: continuous_map X Tx R R_standard_topology u9.
L178982
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L178983
We prove the intermediate claim HI0subR: I0 ⊆ R.
L178985
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L178985
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u9 Hu9contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L178993
Set den9 to be the term mul_SNo den8 den.
L178994
We prove the intermediate claim Hden9R: den9 ∈ R.
L178996
An exact proof term for the current goal is (real_mul_SNo den8 Hden8R den H23R).
L178996
We prove the intermediate claim Hden9pos: 0 < den9.
L178998
We prove the intermediate claim Hden8S: SNo den8.
L178999
An exact proof term for the current goal is (real_SNo den8 Hden8R).
L178999
An exact proof term for the current goal is (mul_SNo_pos_pos den8 den Hden8S H23S Hden8pos HdenPos).
L179000
Set u9s to be the term compose_fun X u9 (mul_const_fun den9).
L179001
We prove the intermediate claim Hu9s_cont: continuous_map X Tx R R_standard_topology u9s.
L179003
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u9 den9 HTx Hu9contR Hden9R Hden9pos).
L179003
Set g9 to be the term compose_fun X (pair_map X g8 u9s) add_fun_R.
L179004
We prove the intermediate claim Hg9cont: continuous_map X Tx R R_standard_topology g9.
L179006
An exact proof term for the current goal is (add_two_continuous_R X Tx g8 u9s HTx Hg8cont Hu9s_cont).
L179006
We prove the intermediate claim Hu9contA: continuous_map A Ta R R_standard_topology u9.
(*** fifteenth correction step scaffold: residual r9 on A and scaling r9s ***)
L179010
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u9 A HTx HAsubX Hu9contR).
L179010
Set u9neg to be the term compose_fun A u9 neg_fun.
L179011
We prove the intermediate claim Hu9neg_cont: continuous_map A Ta R R_standard_topology u9neg.
L179013
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u9 neg_fun Hu9contA Hnegcont).
L179014
We prove the intermediate claim Hr8s_contR: continuous_map A Ta R R_standard_topology r8s.
L179016
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r8s Hr8s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L179017
Set r9 to be the term compose_fun A (pair_map A r8s u9neg) add_fun_R.
L179018
We prove the intermediate claim Hr9_cont: continuous_map A Ta R R_standard_topology r9.
L179020
An exact proof term for the current goal is (add_two_continuous_R A Ta r8s u9neg HTa Hr8s_contR Hu9neg_cont).
L179020
We prove the intermediate claim Hr9_apply: ∀x : set, x ∈ A → apply_fun r9 x = add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x)).
L179023
Let x be given.
L179023
Assume HxA: x ∈ A.
L179023
We prove the intermediate claim Hpimg: apply_fun (pair_map A r8s u9neg) x ∈ setprod R R.
L179025
rewrite the current goal using (pair_map_apply A R R r8s u9neg x HxA) (from left to right).
L179025
We prove the intermediate claim Hr8sxI: apply_fun r8s x ∈ I.
L179027
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r8s Hr8s_cont x HxA).
L179027
We prove the intermediate claim Hr8sxR: apply_fun r8s x ∈ R.
L179029
An exact proof term for the current goal is (HIcR (apply_fun r8s x) Hr8sxI).
L179029
We prove the intermediate claim Hu9negRx: apply_fun u9neg x ∈ R.
L179031
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u9neg Hu9neg_cont x HxA).
L179031
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r8s x) (apply_fun u9neg x) Hr8sxR Hu9negRx).
L179032
rewrite the current goal using (compose_fun_apply A (pair_map A r8s u9neg) add_fun_R x HxA) (from left to right).
L179033
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r8s u9neg) x) Hpimg) (from left to right) at position 1.
L179034
rewrite the current goal using (pair_map_apply A R R r8s u9neg x HxA) (from left to right).
L179035
rewrite the current goal using (tuple_2_0_eq (apply_fun r8s x) (apply_fun u9neg x)) (from left to right).
L179036
rewrite the current goal using (tuple_2_1_eq (apply_fun r8s x) (apply_fun u9neg x)) (from left to right).
L179037
rewrite the current goal using (compose_fun_apply A u9 neg_fun x HxA) (from left to right) at position 1.
L179038
We prove the intermediate claim Hu9Rx: apply_fun u9 x ∈ R.
L179040
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u9 Hu9contA x HxA).
L179040
rewrite the current goal using (neg_fun_apply (apply_fun u9 x) Hu9Rx) (from left to right) at position 1.
Use reflexivity.
L179042
We prove the intermediate claim Hr9_range: ∀x : set, x ∈ A → apply_fun r9 x ∈ I2.
L179044
We prove the intermediate claim Hu9AB: continuous_map X Tx I0 T0 u9 ∧ (∀x0 : set, x0 ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x0 = minus_SNo one_third).
L179048
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u9 ∧ (∀x0 : set, x0 ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r8s ((closed_interval one_third 1) ∩ I) → apply_fun u9 x0 = one_third) Hu9).
L179054
We prove the intermediate claim Hu9_on_B9: ∀x0 : set, x0 ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x0 = minus_SNo one_third.
L179058
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u9) (∀x0 : set, x0 ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x0 = minus_SNo one_third) Hu9AB).
L179062
We prove the intermediate claim Hu9_on_C9: ∀x0 : set, x0 ∈ preimage_of A r8s ((closed_interval one_third 1) ∩ I) → apply_fun u9 x0 = one_third.
L179066
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u9 ∧ (∀x0 : set, x0 ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r8s ((closed_interval one_third 1) ∩ I) → apply_fun u9 x0 = one_third) Hu9).
L179072
Let x be given.
L179073
Assume HxA: x ∈ A.
L179073
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L179074
Set I3 to be the term closed_interval one_third 1.
L179075
Set B9 to be the term preimage_of A r8s (I1 ∩ I).
L179076
Set C9 to be the term preimage_of A r8s (I3 ∩ I).
L179077
We prove the intermediate claim Hr8sIx: apply_fun r8s x ∈ I.
L179079
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r8s Hr8s_cont x HxA).
L179079
We prove the intermediate claim HB9_cases: x ∈ B9 ∨ ¬ (x ∈ B9).
L179081
An exact proof term for the current goal is (xm (x ∈ B9)).
L179081
Apply (HB9_cases (apply_fun r9 x ∈ I2)) to the current goal.
L179083
Assume HxB9: x ∈ B9.
L179083
We prove the intermediate claim Hu9eq: apply_fun u9 x = minus_SNo one_third.
L179085
An exact proof term for the current goal is (Hu9_on_B9 x HxB9).
L179085
We prove the intermediate claim Hr9eq: apply_fun r9 x = add_SNo (apply_fun r8s x) one_third.
L179087
rewrite the current goal using (Hr9_apply x HxA) (from left to right).
L179087
rewrite the current goal using Hu9eq (from left to right) at position 1.
L179088
We prove the intermediate claim H13R: one_third ∈ R.
L179090
An exact proof term for the current goal is one_third_in_R.
L179090
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L179092
rewrite the current goal using Hr9eq (from left to right).
L179093
We prove the intermediate claim Hr8sI1I: apply_fun r8s x ∈ I1 ∩ I.
L179095
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r8s x0 ∈ I1 ∩ I) x HxB9).
L179095
We prove the intermediate claim Hr8sI1: apply_fun r8s x ∈ I1.
L179097
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r8s x) Hr8sI1I).
L179097
We prove the intermediate claim H13R: one_third ∈ R.
L179099
An exact proof term for the current goal is one_third_in_R.
L179099
We prove the intermediate claim H23R: two_thirds ∈ R.
L179101
An exact proof term for the current goal is two_thirds_in_R.
L179101
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L179103
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L179103
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L179105
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L179105
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L179107
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L179107
We prove the intermediate claim Hr8sRx: apply_fun r8s x ∈ R.
L179109
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r8s x) Hr8sI1).
L179109
We prove the intermediate claim Hr8s_bounds: Rle (minus_SNo 1) (apply_fun r8s x) ∧ Rle (apply_fun r8s x) (minus_SNo one_third).
L179111
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r8s x) Hm1R Hm13R Hr8sI1).
L179112
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r8s x).
L179114
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r8s x)) (Rle (apply_fun r8s x) (minus_SNo one_third)) Hr8s_bounds).
L179115
We prove the intermediate claim HhiI1: Rle (apply_fun r8s x) (minus_SNo one_third).
L179117
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r8s x)) (Rle (apply_fun r8s x) (minus_SNo one_third)) Hr8s_bounds).
L179118
We prove the intermediate claim Hr9xR: add_SNo (apply_fun r8s x) one_third ∈ R.
L179120
An exact proof term for the current goal is (real_add_SNo (apply_fun r8s x) Hr8sRx one_third H13R).
L179120
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r8s x) one_third).
L179122
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r8s x) one_third Hm1R Hr8sRx H13R Hm1le).
L179122
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r8s x) one_third).
L179124
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L179124
An exact proof term for the current goal is Hlow_tmp.
L179125
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r8s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L179127
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r8s x) (minus_SNo one_third) one_third Hr8sRx Hm13R H13R HhiI1).
L179127
We prove the intermediate claim H13S: SNo one_third.
L179129
An exact proof term for the current goal is (real_SNo one_third H13R).
L179129
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r8s x) one_third) 0.
L179131
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L179131
An exact proof term for the current goal is Hup0_tmp.
L179132
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L179134
An exact proof term for the current goal is Rle_0_two_thirds.
L179134
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r8s x) one_third) two_thirds.
L179136
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r8s x) one_third) 0 two_thirds Hup0 H0le23).
L179136
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r8s x) one_third) Hm23R H23R Hr9xR Hlow Hup).
L179140
Assume HxnotB9: ¬ (x ∈ B9).
L179140
We prove the intermediate claim HC9_cases: x ∈ C9 ∨ ¬ (x ∈ C9).
L179142
An exact proof term for the current goal is (xm (x ∈ C9)).
L179142
Apply (HC9_cases (apply_fun r9 x ∈ I2)) to the current goal.
L179144
Assume HxC9: x ∈ C9.
L179144
We prove the intermediate claim Hu9eq: apply_fun u9 x = one_third.
L179146
An exact proof term for the current goal is (Hu9_on_C9 x HxC9).
L179146
We prove the intermediate claim Hr9eq: apply_fun r9 x = add_SNo (apply_fun r8s x) (minus_SNo one_third).
L179148
rewrite the current goal using (Hr9_apply x HxA) (from left to right).
L179148
rewrite the current goal using Hu9eq (from left to right) at position 1.
Use reflexivity.
L179150
rewrite the current goal using Hr9eq (from left to right).
L179151
We prove the intermediate claim Hr8sI3I: apply_fun r8s x ∈ I3 ∩ I.
L179153
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r8s x0 ∈ I3 ∩ I) x HxC9).
L179153
We prove the intermediate claim Hr8sI3: apply_fun r8s x ∈ I3.
L179155
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r8s x) Hr8sI3I).
L179155
We prove the intermediate claim H13R: one_third ∈ R.
L179157
An exact proof term for the current goal is one_third_in_R.
L179157
We prove the intermediate claim H23R: two_thirds ∈ R.
L179159
An exact proof term for the current goal is two_thirds_in_R.
L179159
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L179161
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L179161
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L179163
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L179163
We prove the intermediate claim Hr8sRx: apply_fun r8s x ∈ R.
L179165
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r8s x) Hr8sI3).
L179165
We prove the intermediate claim Hr8s_bounds: Rle one_third (apply_fun r8s x) ∧ Rle (apply_fun r8s x) 1.
L179167
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r8s x) H13R real_1 Hr8sI3).
L179167
We prove the intermediate claim HloI3: Rle one_third (apply_fun r8s x).
L179169
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r8s x)) (Rle (apply_fun r8s x) 1) Hr8s_bounds).
L179169
We prove the intermediate claim HhiI3: Rle (apply_fun r8s x) 1.
L179171
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r8s x)) (Rle (apply_fun r8s x) 1) Hr8s_bounds).
L179171
We prove the intermediate claim Hr9xR: add_SNo (apply_fun r8s x) (minus_SNo one_third) ∈ R.
L179173
An exact proof term for the current goal is (real_add_SNo (apply_fun r8s x) Hr8sRx (minus_SNo one_third) Hm13R).
L179173
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r8s x) (minus_SNo one_third)).
L179176
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r8s x) (minus_SNo one_third) H13R Hr8sRx Hm13R HloI3).
L179177
We prove the intermediate claim H13S: SNo one_third.
L179179
An exact proof term for the current goal is (real_SNo one_third H13R).
L179179
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r8s x) (minus_SNo one_third)).
L179181
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L179181
An exact proof term for the current goal is H0le_tmp.
L179182
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L179184
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L179184
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r8s x) (minus_SNo one_third)).
L179186
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r8s x) (minus_SNo one_third)) Hm23le0 H0le).
L179188
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r8s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L179191
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r8s x) 1 (minus_SNo one_third) Hr8sRx real_1 Hm13R HhiI3).
L179192
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L179194
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L179194
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L179195
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r8s x) (minus_SNo one_third)) two_thirds.
L179197
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r8s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L179200
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r8s x) (minus_SNo one_third)) Hm23R H23R Hr9xR Hlow Hup).
L179204
Assume HxnotC9: ¬ (x ∈ C9).
L179204
We prove the intermediate claim HxX: x ∈ X.
L179206
An exact proof term for the current goal is (HAsubX x HxA).
L179206
We prove the intermediate claim HnotI1: ¬ (apply_fun r8s x ∈ I1).
L179208
Assume Hr8sI1': apply_fun r8s x ∈ I1.
L179208
We prove the intermediate claim Hr8sI1I: apply_fun r8s x ∈ I1 ∩ I.
L179210
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r8s x) Hr8sI1' Hr8sIx).
L179210
We prove the intermediate claim HxB9': x ∈ B9.
L179212
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r8s x0 ∈ I1 ∩ I) x HxA Hr8sI1I).
L179212
Apply FalseE to the current goal.
L179213
An exact proof term for the current goal is (HxnotB9 HxB9').
L179214
We prove the intermediate claim HnotI3: ¬ (apply_fun r8s x ∈ I3).
L179216
Assume Hr8sI3': apply_fun r8s x ∈ I3.
L179216
We prove the intermediate claim Hr8sI3I: apply_fun r8s x ∈ I3 ∩ I.
L179218
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r8s x) Hr8sI3' Hr8sIx).
L179218
We prove the intermediate claim HxC9': x ∈ C9.
L179220
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r8s x0 ∈ I3 ∩ I) x HxA Hr8sI3I).
L179220
Apply FalseE to the current goal.
L179221
An exact proof term for the current goal is (HxnotC9 HxC9').
L179222
We prove the intermediate claim Hr8sRx: apply_fun r8s x ∈ R.
L179224
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r8s x) Hr8sIx).
L179224
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L179226
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L179226
We prove the intermediate claim H13R: one_third ∈ R.
L179228
An exact proof term for the current goal is one_third_in_R.
L179228
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L179230
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L179230
We prove the intermediate claim Hr8s_boundsI: Rle (minus_SNo 1) (apply_fun r8s x) ∧ Rle (apply_fun r8s x) 1.
L179232
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r8s x) Hm1R real_1 Hr8sIx).
L179232
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r8s x) (minus_SNo 1)).
L179234
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r8s x) (andEL (Rle (minus_SNo 1) (apply_fun r8s x)) (Rle (apply_fun r8s x) 1) Hr8s_boundsI)).
L179235
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r8s x)).
L179237
An exact proof term for the current goal is (RleE_nlt (apply_fun r8s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r8s x)) (Rle (apply_fun r8s x) 1) Hr8s_boundsI)).
L179238
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r8s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r8s x).
L179240
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r8s x) Hm1R Hm13R Hr8sRx HnotI1).
L179241
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r8s x).
L179243
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r8s x))) to the current goal.
L179244
Assume Hbad: Rlt (apply_fun r8s x) (minus_SNo 1).
L179244
Apply FalseE to the current goal.
L179245
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L179247
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r8s x).
L179247
An exact proof term for the current goal is Hok.
L179248
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r8s x) (minus_SNo one_third)).
L179250
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r8s x) Hm13lt_fx).
L179250
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r8s x) one_third ∨ Rlt 1 (apply_fun r8s x).
L179252
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r8s x) H13R real_1 Hr8sRx HnotI3).
L179253
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r8s x) one_third.
L179255
Apply (HnotI3_cases (Rlt (apply_fun r8s x) one_third)) to the current goal.
L179256
Assume Hok: Rlt (apply_fun r8s x) one_third.
L179256
An exact proof term for the current goal is Hok.
L179258
Assume Hbad: Rlt 1 (apply_fun r8s x).
L179258
Apply FalseE to the current goal.
L179259
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L179260
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r8s x)).
L179262
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r8s x) one_third Hfx_lt_13).
L179262
We prove the intermediate claim Hr8sI0: apply_fun r8s x ∈ I0.
L179264
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L179265
We prove the intermediate claim HxSep: apply_fun r8s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L179267
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r8s x) Hr8sRx (andI (¬ (Rlt (apply_fun r8s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r8s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L179271
rewrite the current goal using HI0_def (from left to right).
L179272
An exact proof term for the current goal is HxSep.
L179273
We prove the intermediate claim Hu9funI0: function_on u9 X I0.
L179275
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u9 Hu9contI0).
L179275
We prove the intermediate claim Hu9xI0: apply_fun u9 x ∈ I0.
L179277
An exact proof term for the current goal is (Hu9funI0 x HxX).
L179277
We prove the intermediate claim Hu9xR: apply_fun u9 x ∈ R.
L179279
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u9 x) Hu9xI0).
L179279
We prove the intermediate claim Hm_u9x_R: minus_SNo (apply_fun u9 x) ∈ R.
L179281
An exact proof term for the current goal is (real_minus_SNo (apply_fun u9 x) Hu9xR).
L179281
rewrite the current goal using (Hr9_apply x HxA) (from left to right).
L179282
We prove the intermediate claim Hr9xR: add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x)) ∈ R.
L179284
An exact proof term for the current goal is (real_add_SNo (apply_fun r8s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r8s x) Hr8sI0) (minus_SNo (apply_fun u9 x)) Hm_u9x_R).
L179286
We prove the intermediate claim H23R: two_thirds ∈ R.
L179288
An exact proof term for the current goal is two_thirds_in_R.
L179288
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L179290
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L179290
We prove the intermediate claim Hr8s_bounds0: Rle (minus_SNo one_third) (apply_fun r8s x) ∧ Rle (apply_fun r8s x) one_third.
L179292
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r8s x) Hm13R H13R Hr8sI0).
L179292
We prove the intermediate claim Hu9_bounds0: Rle (minus_SNo one_third) (apply_fun u9 x) ∧ Rle (apply_fun u9 x) one_third.
L179294
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u9 x) Hm13R H13R Hu9xI0).
L179294
We prove the intermediate claim Hm13_le_r8s: Rle (minus_SNo one_third) (apply_fun r8s x).
L179296
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r8s x)) (Rle (apply_fun r8s x) one_third) Hr8s_bounds0).
L179296
We prove the intermediate claim Hr8s_le_13: Rle (apply_fun r8s x) one_third.
L179298
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r8s x)) (Rle (apply_fun r8s x) one_third) Hr8s_bounds0).
L179298
We prove the intermediate claim Hm13_le_u9x: Rle (minus_SNo one_third) (apply_fun u9 x).
L179300
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u9 x)) (Rle (apply_fun u9 x) one_third) Hu9_bounds0).
L179300
We prove the intermediate claim Hu9x_le_13: Rle (apply_fun u9 x) one_third.
L179302
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u9 x)) (Rle (apply_fun u9 x) one_third) Hu9_bounds0).
L179302
We prove the intermediate claim Hm13_le_mu9: Rle (minus_SNo one_third) (minus_SNo (apply_fun u9 x)).
L179304
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u9 x) one_third Hu9x_le_13).
L179304
We prove the intermediate claim Hmu9_le_13: Rle (minus_SNo (apply_fun u9 x)) one_third.
L179306
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u9 x)) (minus_SNo (minus_SNo one_third)).
L179307
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u9 x) Hm13_le_u9x).
L179307
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L179308
An exact proof term for the current goal is Htmp.
L179309
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u9 x))).
L179312
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u9 x)) Hm13R Hm13R Hm_u9x_R Hm13_le_mu9).
L179313
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u9 x))) (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))).
L179316
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r8s x) (minus_SNo (apply_fun u9 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r8s x) Hr8sI0) Hm_u9x_R Hm13_le_r8s).
L179320
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))).
L179323
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u9 x))) (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))) Hlow1 Hlow2).
L179326
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))).
L179328
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L179328
An exact proof term for the current goal is Hlow_tmp.
L179329
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))) (add_SNo (apply_fun r8s x) one_third).
L179332
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r8s x) (minus_SNo (apply_fun u9 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r8s x) Hr8sI0) Hm_u9x_R H13R Hmu9_le_13).
L179334
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r8s x) one_third) (add_SNo one_third one_third).
L179337
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r8s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r8s x) Hr8sI0) H13R H13R Hr8s_le_13).
L179339
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))) (add_SNo one_third one_third).
L179342
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))) (add_SNo (apply_fun r8s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L179345
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L179347
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))) two_thirds.
L179349
rewrite the current goal using Hdef23 (from left to right) at position 1.
L179349
An exact proof term for the current goal is Hup_tmp.
L179350
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))) Hm23R H23R Hr9xR Hlow Hup).
L179353
Set r9s to be the term compose_fun A r9 (div_const_fun den).
L179354
We prove the intermediate claim Hr9s_cont: continuous_map A Ta I Ti r9s.
L179356
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
L179357
An exact proof term for the current goal is R_standard_topology_is_topology.
L179357
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L179359
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L179359
We prove the intermediate claim Hr9s_contR: continuous_map A Ta R R_standard_topology r9s.
L179361
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r9 (div_const_fun den) Hr9_cont Hdivcont).
L179362
We prove the intermediate claim Hr9s_I: ∀x : set, x ∈ A → apply_fun r9s x ∈ I.
L179364
Let x be given.
L179364
Assume HxA: x ∈ A.
L179364
We prove the intermediate claim Hr9xI2: apply_fun r9 x ∈ I2.
L179366
An exact proof term for the current goal is (Hr9_range x HxA).
L179366
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L179368
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L179368
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L179370
An exact proof term for the current goal is (real_minus_SNo den H23R).
L179370
We prove the intermediate claim Hr9xR: apply_fun r9 x ∈ R.
L179372
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r9 x) Hr9xI2).
L179372
We prove the intermediate claim Hr9xS: SNo (apply_fun r9 x).
L179374
An exact proof term for the current goal is (real_SNo (apply_fun r9 x) Hr9xR).
L179374
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r9 x) ∧ Rle (apply_fun r9 x) den.
L179376
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r9 x) HmdenR H23R Hr9xI2).
L179376
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r9 x).
L179378
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r9 x)) (Rle (apply_fun r9 x) den) Hbounds).
L179380
We prove the intermediate claim Hhi: Rle (apply_fun r9 x) den.
L179382
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r9 x)) (Rle (apply_fun r9 x) den) Hbounds).
L179384
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r9 x)).
L179386
An exact proof term for the current goal is (RleE_nlt (apply_fun r9 x) den Hhi).
L179386
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r9 x) (minus_SNo den)).
L179388
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r9 x) Hlo).
L179388
We prove the intermediate claim HyEq: apply_fun r9s x = div_SNo (apply_fun r9 x) den.
L179390
rewrite the current goal using (compose_fun_apply A r9 (div_const_fun den) x HxA) (from left to right).
L179390
rewrite the current goal using (div_const_fun_apply den (apply_fun r9 x) H23R Hr9xR) (from left to right).
Use reflexivity.
L179392
We prove the intermediate claim HyR: apply_fun r9s x ∈ R.
L179394
rewrite the current goal using HyEq (from left to right).
L179394
An exact proof term for the current goal is (real_div_SNo (apply_fun r9 x) Hr9xR den H23R).
L179395
We prove the intermediate claim HyS: SNo (apply_fun r9s x).
L179397
An exact proof term for the current goal is (real_SNo (apply_fun r9s x) HyR).
L179397
We prove the intermediate claim Hy_le_1: Rle (apply_fun r9s x) 1.
L179399
Apply (RleI (apply_fun r9s x) 1 HyR real_1) to the current goal.
L179399
We will prove ¬ (Rlt 1 (apply_fun r9s x)).
L179400
Assume H1lt: Rlt 1 (apply_fun r9s x).
L179401
We prove the intermediate claim H1lty: 1 < apply_fun r9s x.
L179403
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r9s x) H1lt).
L179403
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r9s x) den.
L179405
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r9s x) den SNo_1 HyS H23S H23pos H1lty).
L179405
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r9s x) den.
L179407
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L179407
An exact proof term for the current goal is HmulLt.
L179408
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r9s x) den = apply_fun r9 x.
L179410
rewrite the current goal using HyEq (from left to right) at position 1.
L179410
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r9 x) den Hr9xS H23S H23ne0).
L179411
We prove the intermediate claim Hden_lt_r9x: den < apply_fun r9 x.
L179413
rewrite the current goal using HmulEq (from right to left).
L179413
An exact proof term for the current goal is HmulLt'.
L179414
We prove the intermediate claim Hbad: Rlt den (apply_fun r9 x).
L179416
An exact proof term for the current goal is (RltI den (apply_fun r9 x) H23R Hr9xR Hden_lt_r9x).
L179416
An exact proof term for the current goal is (Hnlt_hi Hbad).
L179417
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r9s x).
L179419
Apply (RleI (minus_SNo 1) (apply_fun r9s x) Hm1R HyR) to the current goal.
L179419
We will prove ¬ (Rlt (apply_fun r9s x) (minus_SNo 1)).
L179420
Assume Hylt: Rlt (apply_fun r9s x) (minus_SNo 1).
L179421
We prove the intermediate claim Hylts: apply_fun r9s x < minus_SNo 1.
L179423
An exact proof term for the current goal is (RltE_lt (apply_fun r9s x) (minus_SNo 1) Hylt).
L179423
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r9s x) den < mul_SNo (minus_SNo 1) den.
L179425
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r9s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L179426
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r9s x) den = apply_fun r9 x.
L179428
rewrite the current goal using HyEq (from left to right) at position 1.
L179428
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r9 x) den Hr9xS H23S H23ne0).
L179429
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L179431
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L179431
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L179433
We prove the intermediate claim Hr9x_lt_mden: apply_fun r9 x < minus_SNo den.
L179435
rewrite the current goal using HmulEq (from right to left).
L179435
rewrite the current goal using HrhsEq (from right to left).
L179436
An exact proof term for the current goal is HmulLt.
L179437
We prove the intermediate claim Hbad: Rlt (apply_fun r9 x) (minus_SNo den).
L179439
An exact proof term for the current goal is (RltI (apply_fun r9 x) (minus_SNo den) Hr9xR HmdenR Hr9x_lt_mden).
L179439
An exact proof term for the current goal is (Hnlt_lo Hbad).
L179440
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r9s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L179442
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r9s I Hr9s_contR HIcR Hr9s_I).
L179443
We prove the intermediate claim Hex_u10: ∃u10 : set, continuous_map X Tx I0 T0 u10 ∧ (∀x : set, x ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r9s ((closed_interval one_third 1) ∩ I) → apply_fun u10 x = one_third).
(*** further iteration scaffold: build u10 and g10 from residual r9s ***)
L179453
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r9s Hnorm HA Hr9s_cont).
L179453
Apply Hex_u10 to the current goal.
L179454
Let u10 be given.
L179455
Assume Hu10.
L179455
We prove the intermediate claim Hu10contI0: continuous_map X Tx I0 T0 u10.
L179457
We prove the intermediate claim Hu10AB: continuous_map X Tx I0 T0 u10 ∧ (∀x : set, x ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x = minus_SNo one_third).
L179461
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u10 ∧ (∀x : set, x ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r9s ((closed_interval one_third 1) ∩ I) → apply_fun u10 x = one_third) Hu10).
L179467
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u10) (∀x : set, x ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x = minus_SNo one_third) Hu10AB).
L179472
We prove the intermediate claim Hu10contR: continuous_map X Tx R R_standard_topology u10.
L179474
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L179475
We prove the intermediate claim HI0subR: I0 ⊆ R.
L179477
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L179477
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u10 Hu10contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L179485
Set den10 to be the term mul_SNo den9 den.
L179486
We prove the intermediate claim Hden10R: den10 ∈ R.
L179488
An exact proof term for the current goal is (real_mul_SNo den9 Hden9R den H23R).
L179488
We prove the intermediate claim Hden10pos: 0 < den10.
L179490
We prove the intermediate claim Hden9S: SNo den9.
L179491
An exact proof term for the current goal is (real_SNo den9 Hden9R).
L179491
An exact proof term for the current goal is (mul_SNo_pos_pos den9 den Hden9S H23S Hden9pos HdenPos).
L179492
Set u10s to be the term compose_fun X u10 (mul_const_fun den10).
L179493
We prove the intermediate claim Hu10s_cont: continuous_map X Tx R R_standard_topology u10s.
L179495
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u10 den10 HTx Hu10contR Hden10R Hden10pos).
L179495
Set g10 to be the term compose_fun X (pair_map X g9 u10s) add_fun_R.
L179496
We prove the intermediate claim Hg10cont: continuous_map X Tx R R_standard_topology g10.
L179498
An exact proof term for the current goal is (add_two_continuous_R X Tx g9 u10s HTx Hg9cont Hu10s_cont).
L179498
We prove the intermediate claim Hu10contA: continuous_map A Ta R R_standard_topology u10.
(*** further iteration scaffold: residual r10 on A and scaling r10s ***)
L179502
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u10 A HTx HAsubX Hu10contR).
L179502
Set u10neg to be the term compose_fun A u10 neg_fun.
L179503
We prove the intermediate claim Hu10neg_cont: continuous_map A Ta R R_standard_topology u10neg.
L179505
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u10 neg_fun Hu10contA Hnegcont).
L179506
We prove the intermediate claim Hr9s_contR: continuous_map A Ta R R_standard_topology r9s.
L179508
We prove the intermediate claim HTiEq': Ti = subspace_topology R R_standard_topology I.
L179509
An exact proof term for the current goal is HTiEq.
L179509
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r9s Hr9s_cont HIcR R_standard_topology_is_topology_local HTiEq').
L179511
Set r10 to be the term compose_fun A (pair_map A r9s u10neg) add_fun_R.
L179512
We prove the intermediate claim Hr10_cont: continuous_map A Ta R R_standard_topology r10.
L179514
An exact proof term for the current goal is (add_two_continuous_R A Ta r9s u10neg HTa Hr9s_contR Hu10neg_cont).
L179514
We prove the intermediate claim Hr10_apply: ∀x : set, x ∈ A → apply_fun r10 x = add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x)).
L179517
Let x be given.
L179517
Assume HxA: x ∈ A.
L179517
We prove the intermediate claim Hpimg: apply_fun (pair_map A r9s u10neg) x ∈ setprod R R.
L179519
rewrite the current goal using (pair_map_apply A R R r9s u10neg x HxA) (from left to right).
L179519
We prove the intermediate claim Hr9sxI: apply_fun r9s x ∈ I.
L179521
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r9s Hr9s_cont x HxA).
L179521
We prove the intermediate claim Hr9sxR: apply_fun r9s x ∈ R.
L179523
An exact proof term for the current goal is (HIcR (apply_fun r9s x) Hr9sxI).
L179523
We prove the intermediate claim Hu10negRx: apply_fun u10neg x ∈ R.
L179525
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u10neg Hu10neg_cont x HxA).
L179525
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r9s x) (apply_fun u10neg x) Hr9sxR Hu10negRx).
L179526
rewrite the current goal using (compose_fun_apply A (pair_map A r9s u10neg) add_fun_R x HxA) (from left to right).
L179527
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r9s u10neg) x) Hpimg) (from left to right) at position 1.
L179528
rewrite the current goal using (pair_map_apply A R R r9s u10neg x HxA) (from left to right).
L179529
rewrite the current goal using (tuple_2_0_eq (apply_fun r9s x) (apply_fun u10neg x)) (from left to right).
L179530
rewrite the current goal using (tuple_2_1_eq (apply_fun r9s x) (apply_fun u10neg x)) (from left to right).
L179531
rewrite the current goal using (compose_fun_apply A u10 neg_fun x HxA) (from left to right).
L179532
We prove the intermediate claim Hu10Rx: apply_fun u10 x ∈ R.
L179534
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u10 Hu10contA x HxA).
L179534
rewrite the current goal using (neg_fun_apply (apply_fun u10 x) Hu10Rx) (from left to right).
Use reflexivity.
L179536
We prove the intermediate claim Hr10_range: ∀x : set, x ∈ A → apply_fun r10 x ∈ I2.
L179538
We prove the intermediate claim Hu10AB: continuous_map X Tx I0 T0 u10 ∧ (∀x0 : set, x0 ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x0 = minus_SNo one_third).
L179542
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u10 ∧ (∀x0 : set, x0 ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r9s ((closed_interval one_third 1) ∩ I) → apply_fun u10 x0 = one_third) Hu10).
L179548
We prove the intermediate claim Hu10_on_B10: ∀x0 : set, x0 ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x0 = minus_SNo one_third.
L179552
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u10) (∀x0 : set, x0 ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x0 = minus_SNo one_third) Hu10AB).
L179556
We prove the intermediate claim Hu10_on_C10: ∀x0 : set, x0 ∈ preimage_of A r9s ((closed_interval one_third 1) ∩ I) → apply_fun u10 x0 = one_third.
L179560
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u10 ∧ (∀x0 : set, x0 ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r9s ((closed_interval one_third 1) ∩ I) → apply_fun u10 x0 = one_third) Hu10).
L179566
Let x be given.
L179567
Assume HxA: x ∈ A.
L179567
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L179568
Set I3 to be the term closed_interval one_third 1.
L179569
Set B10 to be the term preimage_of A r9s (I1 ∩ I).
L179570
Set C10 to be the term preimage_of A r9s (I3 ∩ I).
L179571
We prove the intermediate claim Hr9sIx: apply_fun r9s x ∈ I.
L179573
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r9s Hr9s_cont x HxA).
L179573
We prove the intermediate claim HB10_cases: x ∈ B10 ∨ ¬ (x ∈ B10).
L179575
An exact proof term for the current goal is (xm (x ∈ B10)).
L179575
Apply (HB10_cases (apply_fun r10 x ∈ I2)) to the current goal.
L179577
Assume HxB10: x ∈ B10.
L179577
We prove the intermediate claim Hu10eq: apply_fun u10 x = minus_SNo one_third.
L179579
An exact proof term for the current goal is (Hu10_on_B10 x HxB10).
L179579
We prove the intermediate claim Hr10eq: apply_fun r10 x = add_SNo (apply_fun r9s x) one_third.
L179581
rewrite the current goal using (Hr10_apply x HxA) (from left to right).
L179581
rewrite the current goal using Hu10eq (from left to right) at position 1.
L179582
We prove the intermediate claim H13R: one_third ∈ R.
L179584
An exact proof term for the current goal is one_third_in_R.
L179584
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L179586
rewrite the current goal using Hr10eq (from left to right).
L179587
We prove the intermediate claim Hr9sI1I: apply_fun r9s x ∈ I1 ∩ I.
L179589
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r9s x0 ∈ I1 ∩ I) x HxB10).
L179589
We prove the intermediate claim Hr9sI1: apply_fun r9s x ∈ I1.
L179591
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r9s x) Hr9sI1I).
L179591
We prove the intermediate claim H13R: one_third ∈ R.
L179593
An exact proof term for the current goal is one_third_in_R.
L179593
We prove the intermediate claim H23R: two_thirds ∈ R.
L179595
An exact proof term for the current goal is two_thirds_in_R.
L179595
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L179597
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L179597
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L179599
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L179599
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L179601
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L179601
We prove the intermediate claim Hr9sRx: apply_fun r9s x ∈ R.
L179603
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r9s x) Hr9sI1).
L179603
We prove the intermediate claim Hr9s_bounds: Rle (minus_SNo 1) (apply_fun r9s x) ∧ Rle (apply_fun r9s x) (minus_SNo one_third).
L179605
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r9s x) Hm1R Hm13R Hr9sI1).
L179606
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r9s x).
L179608
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r9s x)) (Rle (apply_fun r9s x) (minus_SNo one_third)) Hr9s_bounds).
L179609
We prove the intermediate claim HhiI1: Rle (apply_fun r9s x) (minus_SNo one_third).
L179611
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r9s x)) (Rle (apply_fun r9s x) (minus_SNo one_third)) Hr9s_bounds).
L179612
We prove the intermediate claim Hr10xR: add_SNo (apply_fun r9s x) one_third ∈ R.
L179614
An exact proof term for the current goal is (real_add_SNo (apply_fun r9s x) Hr9sRx one_third H13R).
L179614
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r9s x) one_third).
L179616
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r9s x) one_third Hm1R Hr9sRx H13R Hm1le).
L179616
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r9s x) one_third).
L179618
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L179618
An exact proof term for the current goal is Hlow_tmp.
L179619
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r9s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L179621
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r9s x) (minus_SNo one_third) one_third Hr9sRx Hm13R H13R HhiI1).
L179621
We prove the intermediate claim H13S: SNo one_third.
L179623
An exact proof term for the current goal is (real_SNo one_third H13R).
L179623
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r9s x) one_third) 0.
L179625
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L179625
An exact proof term for the current goal is Hup0_tmp.
L179626
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L179628
An exact proof term for the current goal is Rle_0_two_thirds.
L179628
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r9s x) one_third) two_thirds.
L179630
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r9s x) one_third) 0 two_thirds Hup0 H0le23).
L179630
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r9s x) one_third) Hm23R H23R Hr10xR Hlow Hup).
L179634
Assume HxnotB10: ¬ (x ∈ B10).
L179634
We prove the intermediate claim HC10_cases: x ∈ C10 ∨ ¬ (x ∈ C10).
L179636
An exact proof term for the current goal is (xm (x ∈ C10)).
L179636
Apply (HC10_cases (apply_fun r10 x ∈ I2)) to the current goal.
L179638
Assume HxC10: x ∈ C10.
L179638
We prove the intermediate claim Hu10eq: apply_fun u10 x = one_third.
L179640
An exact proof term for the current goal is (Hu10_on_C10 x HxC10).
L179640
We prove the intermediate claim Hr10eq: apply_fun r10 x = add_SNo (apply_fun r9s x) (minus_SNo one_third).
L179642
rewrite the current goal using (Hr10_apply x HxA) (from left to right).
L179642
rewrite the current goal using Hu10eq (from left to right) at position 1.
Use reflexivity.
L179644
rewrite the current goal using Hr10eq (from left to right).
L179645
We prove the intermediate claim Hr9sI3I: apply_fun r9s x ∈ I3 ∩ I.
L179647
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r9s x0 ∈ I3 ∩ I) x HxC10).
L179647
We prove the intermediate claim Hr9sI3: apply_fun r9s x ∈ I3.
L179649
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r9s x) Hr9sI3I).
L179649
We prove the intermediate claim H13R: one_third ∈ R.
L179651
An exact proof term for the current goal is one_third_in_R.
L179651
We prove the intermediate claim H23R: two_thirds ∈ R.
L179653
An exact proof term for the current goal is two_thirds_in_R.
L179653
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L179655
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L179655
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L179657
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L179657
We prove the intermediate claim Hr9sRx: apply_fun r9s x ∈ R.
L179659
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r9s x) Hr9sI3).
L179659
We prove the intermediate claim Hr9s_bounds: Rle one_third (apply_fun r9s x) ∧ Rle (apply_fun r9s x) 1.
L179661
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r9s x) H13R real_1 Hr9sI3).
L179661
We prove the intermediate claim HloI3: Rle one_third (apply_fun r9s x).
L179663
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r9s x)) (Rle (apply_fun r9s x) 1) Hr9s_bounds).
L179663
We prove the intermediate claim HhiI3: Rle (apply_fun r9s x) 1.
L179665
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r9s x)) (Rle (apply_fun r9s x) 1) Hr9s_bounds).
L179665
We prove the intermediate claim Hr10xR: add_SNo (apply_fun r9s x) (minus_SNo one_third) ∈ R.
L179667
An exact proof term for the current goal is (real_add_SNo (apply_fun r9s x) Hr9sRx (minus_SNo one_third) Hm13R).
L179667
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r9s x) (minus_SNo one_third)).
L179670
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r9s x) (minus_SNo one_third) H13R Hr9sRx Hm13R HloI3).
L179671
We prove the intermediate claim H13S: SNo one_third.
L179673
An exact proof term for the current goal is (real_SNo one_third H13R).
L179673
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r9s x) (minus_SNo one_third)).
L179675
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L179675
An exact proof term for the current goal is H0le_tmp.
L179676
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L179678
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L179678
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r9s x) (minus_SNo one_third)).
L179680
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r9s x) (minus_SNo one_third)) Hm23le0 H0le).
L179682
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r9s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L179685
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r9s x) 1 (minus_SNo one_third) Hr9sRx real_1 Hm13R HhiI3).
L179686
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L179688
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L179688
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L179689
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r9s x) (minus_SNo one_third)) two_thirds.
L179691
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r9s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L179694
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r9s x) (minus_SNo one_third)) Hm23R H23R Hr10xR Hlow Hup).
L179698
Assume HxnotC10: ¬ (x ∈ C10).
L179698
We prove the intermediate claim HxX: x ∈ X.
L179700
An exact proof term for the current goal is (HAsubX x HxA).
L179700
We prove the intermediate claim HnotI1: ¬ (apply_fun r9s x ∈ I1).
L179702
Assume Hr9sI1': apply_fun r9s x ∈ I1.
L179702
We prove the intermediate claim Hr9sI1I: apply_fun r9s x ∈ I1 ∩ I.
L179704
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r9s x) Hr9sI1' Hr9sIx).
L179704
We prove the intermediate claim HxB10': x ∈ B10.
L179706
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r9s x0 ∈ I1 ∩ I) x HxA Hr9sI1I).
L179706
Apply FalseE to the current goal.
L179707
An exact proof term for the current goal is (HxnotB10 HxB10').
L179708
We prove the intermediate claim HnotI3: ¬ (apply_fun r9s x ∈ I3).
L179710
Assume Hr9sI3': apply_fun r9s x ∈ I3.
L179710
We prove the intermediate claim Hr9sI3I: apply_fun r9s x ∈ I3 ∩ I.
L179712
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r9s x) Hr9sI3' Hr9sIx).
L179712
We prove the intermediate claim HxC10': x ∈ C10.
L179714
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r9s x0 ∈ I3 ∩ I) x HxA Hr9sI3I).
L179714
Apply FalseE to the current goal.
L179715
An exact proof term for the current goal is (HxnotC10 HxC10').
L179716
We prove the intermediate claim Hr9sRx: apply_fun r9s x ∈ R.
L179718
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r9s x) Hr9sIx).
L179718
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L179720
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L179720
We prove the intermediate claim H13R: one_third ∈ R.
L179722
An exact proof term for the current goal is one_third_in_R.
L179722
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L179724
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L179724
We prove the intermediate claim Hr9s_boundsI: Rle (minus_SNo 1) (apply_fun r9s x) ∧ Rle (apply_fun r9s x) 1.
L179726
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r9s x) Hm1R real_1 Hr9sIx).
L179726
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r9s x) (minus_SNo 1)).
L179728
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r9s x) (andEL (Rle (minus_SNo 1) (apply_fun r9s x)) (Rle (apply_fun r9s x) 1) Hr9s_boundsI)).
L179729
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r9s x)).
L179731
An exact proof term for the current goal is (RleE_nlt (apply_fun r9s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r9s x)) (Rle (apply_fun r9s x) 1) Hr9s_boundsI)).
L179732
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r9s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r9s x).
L179734
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r9s x) Hm1R Hm13R Hr9sRx HnotI1).
L179735
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r9s x).
L179737
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r9s x))) to the current goal.
L179738
Assume Hbad: Rlt (apply_fun r9s x) (minus_SNo 1).
L179738
Apply FalseE to the current goal.
L179739
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L179741
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r9s x).
L179741
An exact proof term for the current goal is Hok.
L179742
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r9s x) (minus_SNo one_third)).
L179744
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r9s x) Hm13lt_fx).
L179744
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r9s x) one_third ∨ Rlt 1 (apply_fun r9s x).
L179746
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r9s x) H13R real_1 Hr9sRx HnotI3).
L179747
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r9s x) one_third.
L179749
Apply (HnotI3_cases (Rlt (apply_fun r9s x) one_third)) to the current goal.
L179750
Assume Hok: Rlt (apply_fun r9s x) one_third.
L179750
An exact proof term for the current goal is Hok.
L179752
Assume Hbad: Rlt 1 (apply_fun r9s x).
L179752
Apply FalseE to the current goal.
L179753
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L179754
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r9s x)).
L179756
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r9s x) one_third Hfx_lt_13).
L179756
We prove the intermediate claim Hr9sI0: apply_fun r9s x ∈ I0.
L179758
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L179759
We prove the intermediate claim HxSep: apply_fun r9s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L179761
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r9s x) Hr9sRx (andI (¬ (Rlt (apply_fun r9s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r9s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L179765
rewrite the current goal using HI0_def (from left to right).
L179766
An exact proof term for the current goal is HxSep.
L179767
We prove the intermediate claim Hu10funI0: function_on u10 X I0.
L179769
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u10 Hu10contI0).
L179769
We prove the intermediate claim Hu10xI0: apply_fun u10 x ∈ I0.
L179771
An exact proof term for the current goal is (Hu10funI0 x HxX).
L179771
We prove the intermediate claim Hu10xR: apply_fun u10 x ∈ R.
L179773
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u10 x) Hu10xI0).
L179773
We prove the intermediate claim Hm_u10x_R: minus_SNo (apply_fun u10 x) ∈ R.
L179775
An exact proof term for the current goal is (real_minus_SNo (apply_fun u10 x) Hu10xR).
L179775
rewrite the current goal using (Hr10_apply x HxA) (from left to right).
L179776
We prove the intermediate claim Hr10xR: add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x)) ∈ R.
L179778
An exact proof term for the current goal is (real_add_SNo (apply_fun r9s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r9s x) Hr9sI0) (minus_SNo (apply_fun u10 x)) Hm_u10x_R).
L179780
We prove the intermediate claim H23R: two_thirds ∈ R.
L179782
An exact proof term for the current goal is two_thirds_in_R.
L179782
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L179784
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L179784
We prove the intermediate claim Hr9s_bounds0: Rle (minus_SNo one_third) (apply_fun r9s x) ∧ Rle (apply_fun r9s x) one_third.
L179786
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r9s x) Hm13R H13R Hr9sI0).
L179786
We prove the intermediate claim Hu10_bounds0: Rle (minus_SNo one_third) (apply_fun u10 x) ∧ Rle (apply_fun u10 x) one_third.
L179788
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u10 x) Hm13R H13R Hu10xI0).
L179788
We prove the intermediate claim Hr9s_le_13: Rle (apply_fun r9s x) one_third.
L179790
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r9s x)) (Rle (apply_fun r9s x) one_third) Hr9s_bounds0).
L179793
We prove the intermediate claim Hr9s_ge_m13: Rle (minus_SNo one_third) (apply_fun r9s x).
L179795
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r9s x)) (Rle (apply_fun r9s x) one_third) Hr9s_bounds0).
L179798
We prove the intermediate claim Hu10_le_13: Rle (apply_fun u10 x) one_third.
L179800
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u10 x)) (Rle (apply_fun u10 x) one_third) Hu10_bounds0).
L179803
We prove the intermediate claim Hu10_ge_m13: Rle (minus_SNo one_third) (apply_fun u10 x).
L179805
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u10 x)) (Rle (apply_fun u10 x) one_third) Hu10_bounds0).
L179808
We prove the intermediate claim Hm13_le_mu10: Rle (minus_SNo one_third) (minus_SNo (apply_fun u10 x)).
L179810
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u10 x) one_third Hu10_le_13).
L179810
We prove the intermediate claim Hmu10_le_13: Rle (minus_SNo (apply_fun u10 x)) one_third.
L179812
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u10 x)) (minus_SNo (minus_SNo one_third)).
L179813
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u10 x) Hu10_ge_m13).
L179813
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L179814
An exact proof term for the current goal is Htmp.
L179815
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u10 x))).
L179818
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u10 x)) Hm13R Hm13R Hm_u10x_R Hm13_le_mu10).
L179819
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u10 x))) (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))).
L179822
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r9s x) (minus_SNo (apply_fun u10 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r9s x) Hr9sI0) Hm_u10x_R Hr9s_ge_m13).
L179825
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))).
L179828
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u10 x))) (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))) Hlow1 Hlow2).
L179831
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))).
L179833
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L179833
An exact proof term for the current goal is Hlow_tmp.
L179834
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))) (add_SNo (apply_fun r9s x) one_third).
L179837
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r9s x) (minus_SNo (apply_fun u10 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r9s x) Hr9sI0) Hm_u10x_R H13R Hmu10_le_13).
L179839
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r9s x) one_third) (add_SNo one_third one_third).
L179842
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r9s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r9s x) Hr9sI0) H13R H13R Hr9s_le_13).
L179844
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))) (add_SNo one_third one_third).
L179847
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))) (add_SNo (apply_fun r9s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L179850
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L179852
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))) two_thirds.
L179854
rewrite the current goal using Hdef23 (from left to right) at position 1.
L179854
An exact proof term for the current goal is Hup_tmp.
L179855
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))) Hm23R H23R Hr10xR Hlow Hup).
L179858
Set r10s to be the term compose_fun A r10 (div_const_fun den).
L179859
We prove the intermediate claim Hr10s_cont: continuous_map A Ta I Ti r10s.
L179861
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
(*** structured: prove continuity into R, then restrict range to I ***)
L179863
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L179863
We prove the intermediate claim Hr10s_contR: continuous_map A Ta R R_standard_topology r10s.
L179865
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r10 (div_const_fun den) Hr10_cont Hdivcont).
L179866
We prove the intermediate claim Hr10s_I: ∀x : set, x ∈ A → apply_fun r10s x ∈ I.
L179868
Let x be given.
L179868
Assume HxA: x ∈ A.
L179868
We prove the intermediate claim Hr10xI2: apply_fun r10 x ∈ I2.
L179870
An exact proof term for the current goal is (Hr10_range x HxA).
L179870
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L179872
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L179872
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L179874
An exact proof term for the current goal is (real_minus_SNo den H23R).
L179874
We prove the intermediate claim Hr10xR: apply_fun r10 x ∈ R.
L179876
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r10 x) Hr10xI2).
L179876
We prove the intermediate claim Hr10xS: SNo (apply_fun r10 x).
L179878
An exact proof term for the current goal is (real_SNo (apply_fun r10 x) Hr10xR).
L179878
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r10 x) ∧ Rle (apply_fun r10 x) den.
L179880
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r10 x) HmdenR H23R Hr10xI2).
L179880
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r10 x).
L179882
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r10 x)) (Rle (apply_fun r10 x) den) Hbounds).
L179884
We prove the intermediate claim Hhi: Rle (apply_fun r10 x) den.
L179886
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r10 x)) (Rle (apply_fun r10 x) den) Hbounds).
L179888
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r10 x)).
L179890
An exact proof term for the current goal is (RleE_nlt (apply_fun r10 x) den Hhi).
L179890
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r10 x) (minus_SNo den)).
L179892
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r10 x) Hlo).
L179892
We prove the intermediate claim HyEq: apply_fun r10s x = div_SNo (apply_fun r10 x) den.
L179894
rewrite the current goal using (compose_fun_apply A r10 (div_const_fun den) x HxA) (from left to right).
L179894
rewrite the current goal using (div_const_fun_apply den (apply_fun r10 x) H23R Hr10xR) (from left to right).
Use reflexivity.
L179896
We prove the intermediate claim HyR: apply_fun r10s x ∈ R.
L179898
rewrite the current goal using HyEq (from left to right).
L179898
An exact proof term for the current goal is (real_div_SNo (apply_fun r10 x) Hr10xR den H23R).
L179899
We prove the intermediate claim HyS: SNo (apply_fun r10s x).
L179901
An exact proof term for the current goal is (real_SNo (apply_fun r10s x) HyR).
L179901
We prove the intermediate claim Hy_le_1: Rle (apply_fun r10s x) 1.
L179903
Apply (RleI (apply_fun r10s x) 1 HyR real_1) to the current goal.
L179903
We will prove ¬ (Rlt 1 (apply_fun r10s x)).
L179904
Assume H1lt: Rlt 1 (apply_fun r10s x).
L179905
We prove the intermediate claim H1lty: 1 < apply_fun r10s x.
L179907
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r10s x) H1lt).
L179907
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r10s x) den.
L179909
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r10s x) den SNo_1 HyS H23S H23pos H1lty).
L179909
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r10s x) den.
L179911
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L179911
An exact proof term for the current goal is HmulLt.
L179912
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r10s x) den = apply_fun r10 x.
L179914
rewrite the current goal using HyEq (from left to right) at position 1.
L179914
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r10 x) den Hr10xS H23S H23ne0).
L179915
We prove the intermediate claim Hden_lt_r10x: den < apply_fun r10 x.
L179917
rewrite the current goal using HmulEq (from right to left).
L179917
An exact proof term for the current goal is HmulLt'.
L179918
We prove the intermediate claim Hbad: Rlt den (apply_fun r10 x).
L179920
An exact proof term for the current goal is (RltI den (apply_fun r10 x) H23R Hr10xR Hden_lt_r10x).
L179920
An exact proof term for the current goal is (Hnlt_hi Hbad).
L179921
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r10s x).
L179923
Apply (RleI (minus_SNo 1) (apply_fun r10s x) Hm1R HyR) to the current goal.
L179923
We will prove ¬ (Rlt (apply_fun r10s x) (minus_SNo 1)).
L179924
Assume Hylt: Rlt (apply_fun r10s x) (minus_SNo 1).
L179925
We prove the intermediate claim Hylts: apply_fun r10s x < minus_SNo 1.
L179927
An exact proof term for the current goal is (RltE_lt (apply_fun r10s x) (minus_SNo 1) Hylt).
L179927
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r10s x) den < mul_SNo (minus_SNo 1) den.
L179929
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r10s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L179930
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r10s x) den = apply_fun r10 x.
L179932
rewrite the current goal using HyEq (from left to right) at position 1.
L179932
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r10 x) den Hr10xS H23S H23ne0).
L179933
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L179935
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L179935
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L179937
We prove the intermediate claim Hr10x_lt_mden: apply_fun r10 x < minus_SNo den.
L179939
rewrite the current goal using HmulEq (from right to left).
L179939
rewrite the current goal using HrhsEq (from right to left).
L179940
An exact proof term for the current goal is HmulLt.
L179941
We prove the intermediate claim Hbad: Rlt (apply_fun r10 x) (minus_SNo den).
L179943
An exact proof term for the current goal is (RltI (apply_fun r10 x) (minus_SNo den) Hr10xR HmdenR Hr10x_lt_mden).
L179943
An exact proof term for the current goal is (Hnlt_lo Hbad).
L179944
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r10s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L179946
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r10s I Hr10s_contR HIcR Hr10s_I).
L179948
We prove the intermediate claim Hex_u11: ∃u11 : set, continuous_map X Tx I0 T0 u11 ∧ (∀x : set, x ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r10s ((closed_interval one_third 1) ∩ I) → apply_fun u11 x = one_third).
(*** further iteration scaffold: build u11 and g11 from residual r10s ***)
L179958
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r10s Hnorm HA Hr10s_cont).
L179958
Apply Hex_u11 to the current goal.
L179959
Let u11 be given.
L179960
Assume Hu11.
L179960
We prove the intermediate claim Hu11contI0: continuous_map X Tx I0 T0 u11.
L179962
We prove the intermediate claim Hu11AB: continuous_map X Tx I0 T0 u11 ∧ (∀x : set, x ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x = minus_SNo one_third).
L179966
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u11 ∧ (∀x : set, x ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r10s ((closed_interval one_third 1) ∩ I) → apply_fun u11 x = one_third) Hu11).
L179972
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u11) (∀x : set, x ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x = minus_SNo one_third) Hu11AB).
L179977
We prove the intermediate claim Hu11contR: continuous_map X Tx R R_standard_topology u11.
L179979
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L179980
We prove the intermediate claim HI0subR: I0 ⊆ R.
L179982
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L179982
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u11 Hu11contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L179990
Set den11 to be the term mul_SNo den10 den.
L179991
We prove the intermediate claim Hden11R: den11 ∈ R.
L179993
An exact proof term for the current goal is (real_mul_SNo den10 Hden10R den H23R).
L179993
We prove the intermediate claim Hden11pos: 0 < den11.
L179995
We prove the intermediate claim Hden10S: SNo den10.
L179996
An exact proof term for the current goal is (real_SNo den10 Hden10R).
L179996
An exact proof term for the current goal is (mul_SNo_pos_pos den10 den Hden10S H23S Hden10pos HdenPos).
L179997
Set u11s to be the term compose_fun X u11 (mul_const_fun den11).
L179998
We prove the intermediate claim Hu11s_cont: continuous_map X Tx R R_standard_topology u11s.
L180000
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u11 den11 HTx Hu11contR Hden11R Hden11pos).
L180000
Set g11 to be the term compose_fun X (pair_map X g10 u11s) add_fun_R.
L180001
We prove the intermediate claim Hg11cont: continuous_map X Tx R R_standard_topology g11.
L180003
An exact proof term for the current goal is (add_two_continuous_R X Tx g10 u11s HTx Hg10cont Hu11s_cont).
L180003
We prove the intermediate claim Hu11contA: continuous_map A Ta R R_standard_topology u11.
(*** further iteration scaffold: residual r11 on A and scaling r11s ***)
L180007
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u11 A HTx HAsubX Hu11contR).
L180007
Set u11neg to be the term compose_fun A u11 neg_fun.
L180008
We prove the intermediate claim Hu11neg_cont: continuous_map A Ta R R_standard_topology u11neg.
L180010
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u11 neg_fun Hu11contA Hnegcont).
L180011
We prove the intermediate claim Hr10s_contR: continuous_map A Ta R R_standard_topology r10s.
L180013
We prove the intermediate claim HTiEq'': Ti = subspace_topology R R_standard_topology I.
L180014
An exact proof term for the current goal is HTiEq.
L180014
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r10s Hr10s_cont HIcR R_standard_topology_is_topology_local HTiEq'').
L180016
Set r11 to be the term compose_fun A (pair_map A r10s u11neg) add_fun_R.
L180017
We prove the intermediate claim Hr11_cont: continuous_map A Ta R R_standard_topology r11.
L180019
An exact proof term for the current goal is (add_two_continuous_R A Ta r10s u11neg HTa Hr10s_contR Hu11neg_cont).
L180019
We prove the intermediate claim Hr11_apply: ∀x : set, x ∈ A → apply_fun r11 x = add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x)).
L180022
Let x be given.
L180022
Assume HxA: x ∈ A.
L180022
We prove the intermediate claim Hpimg: apply_fun (pair_map A r10s u11neg) x ∈ setprod R R.
L180024
rewrite the current goal using (pair_map_apply A R R r10s u11neg x HxA) (from left to right).
L180024
We prove the intermediate claim Hr10sxI: apply_fun r10s x ∈ I.
L180026
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r10s Hr10s_cont x HxA).
L180026
We prove the intermediate claim Hr10sxR: apply_fun r10s x ∈ R.
L180028
An exact proof term for the current goal is (HIcR (apply_fun r10s x) Hr10sxI).
L180028
We prove the intermediate claim Hu11negRx: apply_fun u11neg x ∈ R.
L180030
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u11neg Hu11neg_cont x HxA).
L180030
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r10s x) (apply_fun u11neg x) Hr10sxR Hu11negRx).
L180031
rewrite the current goal using (compose_fun_apply A (pair_map A r10s u11neg) add_fun_R x HxA) (from left to right).
L180032
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r10s u11neg) x) Hpimg) (from left to right) at position 1.
L180033
rewrite the current goal using (pair_map_apply A R R r10s u11neg x HxA) (from left to right).
L180034
rewrite the current goal using (tuple_2_0_eq (apply_fun r10s x) (apply_fun u11neg x)) (from left to right).
L180035
rewrite the current goal using (tuple_2_1_eq (apply_fun r10s x) (apply_fun u11neg x)) (from left to right).
L180036
rewrite the current goal using (compose_fun_apply A u11 neg_fun x HxA) (from left to right) at position 1.
L180037
We prove the intermediate claim Hu11Rx: apply_fun u11 x ∈ R.
L180039
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u11 Hu11contA x HxA).
L180039
rewrite the current goal using (neg_fun_apply (apply_fun u11 x) Hu11Rx) (from left to right) at position 1.
Use reflexivity.
L180041
We prove the intermediate claim Hr11_range: ∀x : set, x ∈ A → apply_fun r11 x ∈ I2.
L180043
We prove the intermediate claim Hu11AB: continuous_map X Tx I0 T0 u11 ∧ (∀x0 : set, x0 ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x0 = minus_SNo one_third).
L180047
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u11 ∧ (∀x0 : set, x0 ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r10s ((closed_interval one_third 1) ∩ I) → apply_fun u11 x0 = one_third) Hu11).
L180053
We prove the intermediate claim Hu11_on_B11: ∀x0 : set, x0 ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x0 = minus_SNo one_third.
L180057
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u11) (∀x0 : set, x0 ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x0 = minus_SNo one_third) Hu11AB).
L180061
We prove the intermediate claim Hu11_on_C11: ∀x0 : set, x0 ∈ preimage_of A r10s ((closed_interval one_third 1) ∩ I) → apply_fun u11 x0 = one_third.
L180065
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u11 ∧ (∀x0 : set, x0 ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r10s ((closed_interval one_third 1) ∩ I) → apply_fun u11 x0 = one_third) Hu11).
L180071
Let x be given.
L180072
Assume HxA: x ∈ A.
L180072
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L180073
Set I3 to be the term closed_interval one_third 1.
L180074
Set B11 to be the term preimage_of A r10s (I1 ∩ I).
L180075
Set C11 to be the term preimage_of A r10s (I3 ∩ I).
L180076
We prove the intermediate claim Hr10sIx: apply_fun r10s x ∈ I.
L180078
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r10s Hr10s_cont x HxA).
L180078
We prove the intermediate claim HB11_cases: x ∈ B11 ∨ ¬ (x ∈ B11).
L180080
An exact proof term for the current goal is (xm (x ∈ B11)).
L180080
Apply (HB11_cases (apply_fun r11 x ∈ I2)) to the current goal.
L180082
Assume HxB11: x ∈ B11.
L180082
We prove the intermediate claim Hu11eq: apply_fun u11 x = minus_SNo one_third.
L180084
An exact proof term for the current goal is (Hu11_on_B11 x HxB11).
L180084
We prove the intermediate claim Hr11eq: apply_fun r11 x = add_SNo (apply_fun r10s x) one_third.
L180086
rewrite the current goal using (Hr11_apply x HxA) (from left to right).
L180086
rewrite the current goal using Hu11eq (from left to right) at position 1.
L180087
We prove the intermediate claim H13R: one_third ∈ R.
L180089
An exact proof term for the current goal is one_third_in_R.
L180089
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L180091
rewrite the current goal using Hr11eq (from left to right).
L180092
We prove the intermediate claim Hr10sI1I: apply_fun r10s x ∈ I1 ∩ I.
L180094
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r10s x0 ∈ I1 ∩ I) x HxB11).
L180094
We prove the intermediate claim Hr10sI1: apply_fun r10s x ∈ I1.
L180096
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r10s x) Hr10sI1I).
L180096
We prove the intermediate claim H13R: one_third ∈ R.
L180098
An exact proof term for the current goal is one_third_in_R.
L180098
We prove the intermediate claim H23R: two_thirds ∈ R.
L180100
An exact proof term for the current goal is two_thirds_in_R.
L180100
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L180102
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L180102
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L180104
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L180104
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L180106
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L180106
We prove the intermediate claim Hr10sRx: apply_fun r10s x ∈ R.
L180108
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r10s x) Hr10sI1).
L180108
We prove the intermediate claim Hr10s_bounds: Rle (minus_SNo 1) (apply_fun r10s x) ∧ Rle (apply_fun r10s x) (minus_SNo one_third).
L180110
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r10s x) Hm1R Hm13R Hr10sI1).
L180111
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r10s x).
L180113
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r10s x)) (Rle (apply_fun r10s x) (minus_SNo one_third)) Hr10s_bounds).
L180114
We prove the intermediate claim HhiI1: Rle (apply_fun r10s x) (minus_SNo one_third).
L180116
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r10s x)) (Rle (apply_fun r10s x) (minus_SNo one_third)) Hr10s_bounds).
L180117
We prove the intermediate claim Hr11xR: add_SNo (apply_fun r10s x) one_third ∈ R.
L180119
An exact proof term for the current goal is (real_add_SNo (apply_fun r10s x) Hr10sRx one_third H13R).
L180119
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r10s x) one_third).
L180121
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r10s x) one_third Hm1R Hr10sRx H13R Hm1le).
L180121
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r10s x) one_third).
L180123
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L180123
An exact proof term for the current goal is Hlow_tmp.
L180124
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r10s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L180126
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r10s x) (minus_SNo one_third) one_third Hr10sRx Hm13R H13R HhiI1).
L180126
We prove the intermediate claim H13S: SNo one_third.
L180128
An exact proof term for the current goal is (real_SNo one_third H13R).
L180128
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r10s x) one_third) 0.
L180130
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L180130
An exact proof term for the current goal is Hup0_tmp.
L180131
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L180133
An exact proof term for the current goal is Rle_0_two_thirds.
L180133
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r10s x) one_third) two_thirds.
L180135
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r10s x) one_third) 0 two_thirds Hup0 H0le23).
L180135
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r10s x) one_third) Hm23R H23R Hr11xR Hlow Hup).
L180139
Assume HxnotB11: ¬ (x ∈ B11).
L180139
We prove the intermediate claim HC11_cases: x ∈ C11 ∨ ¬ (x ∈ C11).
L180141
An exact proof term for the current goal is (xm (x ∈ C11)).
L180141
Apply (HC11_cases (apply_fun r11 x ∈ I2)) to the current goal.
L180143
Assume HxC11: x ∈ C11.
L180143
We prove the intermediate claim Hu11eq: apply_fun u11 x = one_third.
L180145
An exact proof term for the current goal is (Hu11_on_C11 x HxC11).
L180145
We prove the intermediate claim Hr11eq: apply_fun r11 x = add_SNo (apply_fun r10s x) (minus_SNo one_third).
L180147
rewrite the current goal using (Hr11_apply x HxA) (from left to right).
L180147
rewrite the current goal using Hu11eq (from left to right) at position 1.
Use reflexivity.
L180149
rewrite the current goal using Hr11eq (from left to right).
L180150
We prove the intermediate claim Hr10sI3I: apply_fun r10s x ∈ I3 ∩ I.
L180152
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r10s x0 ∈ I3 ∩ I) x HxC11).
L180152
We prove the intermediate claim Hr10sI3: apply_fun r10s x ∈ I3.
L180154
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r10s x) Hr10sI3I).
L180154
We prove the intermediate claim H13R: one_third ∈ R.
L180156
An exact proof term for the current goal is one_third_in_R.
L180156
We prove the intermediate claim H23R: two_thirds ∈ R.
L180158
An exact proof term for the current goal is two_thirds_in_R.
L180158
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L180160
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L180160
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L180162
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L180162
We prove the intermediate claim Hr10sRx: apply_fun r10s x ∈ R.
L180164
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r10s x) Hr10sI3).
L180164
We prove the intermediate claim Hr10s_bounds: Rle one_third (apply_fun r10s x) ∧ Rle (apply_fun r10s x) 1.
L180166
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r10s x) H13R real_1 Hr10sI3).
L180166
We prove the intermediate claim HloI3: Rle one_third (apply_fun r10s x).
L180168
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r10s x)) (Rle (apply_fun r10s x) 1) Hr10s_bounds).
L180168
We prove the intermediate claim HhiI3: Rle (apply_fun r10s x) 1.
L180170
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r10s x)) (Rle (apply_fun r10s x) 1) Hr10s_bounds).
L180170
We prove the intermediate claim Hr11xR: add_SNo (apply_fun r10s x) (minus_SNo one_third) ∈ R.
L180172
An exact proof term for the current goal is (real_add_SNo (apply_fun r10s x) Hr10sRx (minus_SNo one_third) Hm13R).
L180172
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r10s x) (minus_SNo one_third)).
L180175
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r10s x) (minus_SNo one_third) H13R Hr10sRx Hm13R HloI3).
L180176
We prove the intermediate claim H13S: SNo one_third.
L180178
An exact proof term for the current goal is (real_SNo one_third H13R).
L180178
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r10s x) (minus_SNo one_third)).
L180180
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L180180
An exact proof term for the current goal is H0le_tmp.
L180181
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L180183
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L180183
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r10s x) (minus_SNo one_third)).
L180185
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r10s x) (minus_SNo one_third)) Hm23le0 H0le).
L180187
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r10s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L180190
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r10s x) 1 (minus_SNo one_third) Hr10sRx real_1 Hm13R HhiI3).
L180191
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L180193
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L180193
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L180194
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r10s x) (minus_SNo one_third)) two_thirds.
L180196
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r10s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L180199
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r10s x) (minus_SNo one_third)) Hm23R H23R Hr11xR Hlow Hup).
L180203
Assume HxnotC11: ¬ (x ∈ C11).
L180203
We prove the intermediate claim HxX: x ∈ X.
L180205
An exact proof term for the current goal is (HAsubX x HxA).
L180205
We prove the intermediate claim HnotI1: ¬ (apply_fun r10s x ∈ I1).
L180207
Assume Hr10sI1': apply_fun r10s x ∈ I1.
L180207
We prove the intermediate claim Hr10sI1I: apply_fun r10s x ∈ I1 ∩ I.
L180209
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r10s x) Hr10sI1' Hr10sIx).
L180209
We prove the intermediate claim HxB11': x ∈ B11.
L180211
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r10s x0 ∈ I1 ∩ I) x HxA Hr10sI1I).
L180211
Apply FalseE to the current goal.
L180212
An exact proof term for the current goal is (HxnotB11 HxB11').
L180213
We prove the intermediate claim HnotI3: ¬ (apply_fun r10s x ∈ I3).
L180215
Assume Hr10sI3': apply_fun r10s x ∈ I3.
L180215
We prove the intermediate claim Hr10sI3I: apply_fun r10s x ∈ I3 ∩ I.
L180217
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r10s x) Hr10sI3' Hr10sIx).
L180217
We prove the intermediate claim HxC11': x ∈ C11.
L180219
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r10s x0 ∈ I3 ∩ I) x HxA Hr10sI3I).
L180219
Apply FalseE to the current goal.
L180220
An exact proof term for the current goal is (HxnotC11 HxC11').
L180221
We prove the intermediate claim Hr10sRx: apply_fun r10s x ∈ R.
L180223
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r10s x) Hr10sIx).
L180223
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L180225
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L180225
We prove the intermediate claim H13R: one_third ∈ R.
L180227
An exact proof term for the current goal is one_third_in_R.
L180227
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L180229
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L180229
We prove the intermediate claim Hr10s_boundsI: Rle (minus_SNo 1) (apply_fun r10s x) ∧ Rle (apply_fun r10s x) 1.
L180231
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r10s x) Hm1R real_1 Hr10sIx).
L180231
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r10s x) (minus_SNo 1)).
L180233
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r10s x) (andEL (Rle (minus_SNo 1) (apply_fun r10s x)) (Rle (apply_fun r10s x) 1) Hr10s_boundsI)).
L180234
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r10s x)).
L180236
An exact proof term for the current goal is (RleE_nlt (apply_fun r10s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r10s x)) (Rle (apply_fun r10s x) 1) Hr10s_boundsI)).
L180237
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r10s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r10s x).
L180239
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r10s x) Hm1R Hm13R Hr10sRx HnotI1).
L180240
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r10s x).
L180242
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r10s x))) to the current goal.
L180243
Assume Hbad: Rlt (apply_fun r10s x) (minus_SNo 1).
L180243
Apply FalseE to the current goal.
L180244
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L180246
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r10s x).
L180246
An exact proof term for the current goal is Hok.
L180247
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r10s x) (minus_SNo one_third)).
L180249
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r10s x) Hm13lt_fx).
L180249
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r10s x) one_third ∨ Rlt 1 (apply_fun r10s x).
L180251
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r10s x) H13R real_1 Hr10sRx HnotI3).
L180252
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r10s x) one_third.
L180254
Apply (HnotI3_cases (Rlt (apply_fun r10s x) one_third)) to the current goal.
L180255
Assume Hok: Rlt (apply_fun r10s x) one_third.
L180255
An exact proof term for the current goal is Hok.
L180257
Assume Hbad: Rlt 1 (apply_fun r10s x).
L180257
Apply FalseE to the current goal.
L180258
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L180259
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r10s x)).
L180261
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r10s x) one_third Hfx_lt_13).
L180261
We prove the intermediate claim Hr10sI0: apply_fun r10s x ∈ I0.
L180263
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L180264
We prove the intermediate claim HxSep: apply_fun r10s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L180266
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r10s x) Hr10sRx (andI (¬ (Rlt (apply_fun r10s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r10s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L180270
rewrite the current goal using HI0_def (from left to right).
L180271
An exact proof term for the current goal is HxSep.
L180272
We prove the intermediate claim Hu11funI0: function_on u11 X I0.
L180274
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u11 Hu11contI0).
L180274
We prove the intermediate claim Hu11xI0: apply_fun u11 x ∈ I0.
L180276
An exact proof term for the current goal is (Hu11funI0 x HxX).
L180276
We prove the intermediate claim Hu11xR: apply_fun u11 x ∈ R.
L180278
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u11 x) Hu11xI0).
L180278
We prove the intermediate claim Hm_u11x_R: minus_SNo (apply_fun u11 x) ∈ R.
L180280
An exact proof term for the current goal is (real_minus_SNo (apply_fun u11 x) Hu11xR).
L180280
rewrite the current goal using (Hr11_apply x HxA) (from left to right).
L180281
We prove the intermediate claim Hr11xR: add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x)) ∈ R.
L180283
An exact proof term for the current goal is (real_add_SNo (apply_fun r10s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r10s x) Hr10sI0) (minus_SNo (apply_fun u11 x)) Hm_u11x_R).
L180285
We prove the intermediate claim H23R: two_thirds ∈ R.
L180287
An exact proof term for the current goal is two_thirds_in_R.
L180287
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L180289
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L180289
We prove the intermediate claim Hr10s_bounds0: Rle (minus_SNo one_third) (apply_fun r10s x) ∧ Rle (apply_fun r10s x) one_third.
L180291
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r10s x) Hm13R H13R Hr10sI0).
L180291
We prove the intermediate claim Hu11_bounds0: Rle (minus_SNo one_third) (apply_fun u11 x) ∧ Rle (apply_fun u11 x) one_third.
L180293
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u11 x) Hm13R H13R Hu11xI0).
L180293
We prove the intermediate claim Hr10s_le_13: Rle (apply_fun r10s x) one_third.
L180295
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r10s x)) (Rle (apply_fun r10s x) one_third) Hr10s_bounds0).
L180298
We prove the intermediate claim Hr10s_ge_m13: Rle (minus_SNo one_third) (apply_fun r10s x).
L180300
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r10s x)) (Rle (apply_fun r10s x) one_third) Hr10s_bounds0).
L180303
We prove the intermediate claim Hu11_le_13: Rle (apply_fun u11 x) one_third.
L180305
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u11 x)) (Rle (apply_fun u11 x) one_third) Hu11_bounds0).
L180308
We prove the intermediate claim Hu11_ge_m13: Rle (minus_SNo one_third) (apply_fun u11 x).
L180310
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u11 x)) (Rle (apply_fun u11 x) one_third) Hu11_bounds0).
L180313
We prove the intermediate claim Hm13_le_mu11: Rle (minus_SNo one_third) (minus_SNo (apply_fun u11 x)).
L180315
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u11 x) one_third Hu11_le_13).
L180315
We prove the intermediate claim Hmu11_le_13: Rle (minus_SNo (apply_fun u11 x)) one_third.
L180317
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u11 x)) (minus_SNo (minus_SNo one_third)).
L180318
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u11 x) Hu11_ge_m13).
L180318
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L180319
An exact proof term for the current goal is Htmp.
L180320
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u11 x))).
L180323
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u11 x)) Hm13R Hm13R Hm_u11x_R Hm13_le_mu11).
L180324
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u11 x))) (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))).
L180327
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r10s x) (minus_SNo (apply_fun u11 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r10s x) Hr10sI0) Hm_u11x_R Hr10s_ge_m13).
L180330
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))).
L180333
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u11 x))) (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))) Hlow1 Hlow2).
L180336
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))).
L180338
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L180338
An exact proof term for the current goal is Hlow_tmp.
L180339
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))) (add_SNo (apply_fun r10s x) one_third).
L180342
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r10s x) (minus_SNo (apply_fun u11 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r10s x) Hr10sI0) Hm_u11x_R H13R Hmu11_le_13).
L180344
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r10s x) one_third) (add_SNo one_third one_third).
L180347
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r10s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r10s x) Hr10sI0) H13R H13R Hr10s_le_13).
L180349
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))) (add_SNo one_third one_third).
L180352
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))) (add_SNo (apply_fun r10s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L180355
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L180357
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))) two_thirds.
L180359
rewrite the current goal using Hdef23 (from left to right) at position 1.
L180359
An exact proof term for the current goal is Hup_tmp.
L180360
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))) Hm23R H23R Hr11xR Hlow Hup).
L180363
Set r11s to be the term compose_fun A r11 (div_const_fun den).
L180364
We prove the intermediate claim Hr11s_cont: continuous_map A Ta I Ti r11s.
L180366
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
(*** structured: prove continuity into R, then restrict range to I ***)
L180368
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L180368
We prove the intermediate claim Hr11s_contR: continuous_map A Ta R R_standard_topology r11s.
L180370
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r11 (div_const_fun den) Hr11_cont Hdivcont).
L180371
We prove the intermediate claim Hr11s_I: ∀x : set, x ∈ A → apply_fun r11s x ∈ I.
L180373
Let x be given.
L180373
Assume HxA: x ∈ A.
L180373
We prove the intermediate claim Hr11xI2: apply_fun r11 x ∈ I2.
L180375
An exact proof term for the current goal is (Hr11_range x HxA).
L180375
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L180377
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L180377
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L180379
An exact proof term for the current goal is (real_minus_SNo den H23R).
L180379
We prove the intermediate claim Hr11xR: apply_fun r11 x ∈ R.
L180381
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r11 x) Hr11xI2).
L180381
We prove the intermediate claim Hr11xS: SNo (apply_fun r11 x).
L180383
An exact proof term for the current goal is (real_SNo (apply_fun r11 x) Hr11xR).
L180383
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r11 x) ∧ Rle (apply_fun r11 x) den.
L180385
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r11 x) HmdenR H23R Hr11xI2).
L180385
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r11 x).
L180387
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r11 x)) (Rle (apply_fun r11 x) den) Hbounds).
L180389
We prove the intermediate claim Hhi: Rle (apply_fun r11 x) den.
L180391
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r11 x)) (Rle (apply_fun r11 x) den) Hbounds).
L180393
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r11 x)).
L180395
An exact proof term for the current goal is (RleE_nlt (apply_fun r11 x) den Hhi).
L180395
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r11 x) (minus_SNo den)).
L180397
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r11 x) Hlo).
L180397
We prove the intermediate claim HyEq: apply_fun r11s x = div_SNo (apply_fun r11 x) den.
L180399
rewrite the current goal using (compose_fun_apply A r11 (div_const_fun den) x HxA) (from left to right).
L180399
rewrite the current goal using (div_const_fun_apply den (apply_fun r11 x) H23R Hr11xR) (from left to right).
Use reflexivity.
L180401
We prove the intermediate claim HyR: apply_fun r11s x ∈ R.
L180403
rewrite the current goal using HyEq (from left to right).
L180403
An exact proof term for the current goal is (real_div_SNo (apply_fun r11 x) Hr11xR den H23R).
L180404
We prove the intermediate claim HyS: SNo (apply_fun r11s x).
L180406
An exact proof term for the current goal is (real_SNo (apply_fun r11s x) HyR).
L180406
We prove the intermediate claim Hy_le_1: Rle (apply_fun r11s x) 1.
L180408
Apply (RleI (apply_fun r11s x) 1 HyR real_1) to the current goal.
L180408
We will prove ¬ (Rlt 1 (apply_fun r11s x)).
L180409
Assume H1lt: Rlt 1 (apply_fun r11s x).
L180410
We prove the intermediate claim H1lty: 1 < apply_fun r11s x.
L180412
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r11s x) H1lt).
L180412
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r11s x) den.
L180414
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r11s x) den SNo_1 HyS H23S H23pos H1lty).
L180414
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r11s x) den.
L180416
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L180416
An exact proof term for the current goal is HmulLt.
L180417
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r11s x) den = apply_fun r11 x.
L180419
rewrite the current goal using HyEq (from left to right) at position 1.
L180419
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r11 x) den Hr11xS H23S H23ne0).
L180420
We prove the intermediate claim Hden_lt_r11x: den < apply_fun r11 x.
L180422
rewrite the current goal using HmulEq (from right to left).
L180422
An exact proof term for the current goal is HmulLt'.
L180423
We prove the intermediate claim Hbad: Rlt den (apply_fun r11 x).
L180425
An exact proof term for the current goal is (RltI den (apply_fun r11 x) H23R Hr11xR Hden_lt_r11x).
L180425
An exact proof term for the current goal is (Hnlt_hi Hbad).
L180426
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r11s x).
L180428
Apply (RleI (minus_SNo 1) (apply_fun r11s x) Hm1R HyR) to the current goal.
L180428
We will prove ¬ (Rlt (apply_fun r11s x) (minus_SNo 1)).
L180429
Assume Hylt: Rlt (apply_fun r11s x) (minus_SNo 1).
L180430
We prove the intermediate claim Hylts: apply_fun r11s x < minus_SNo 1.
L180432
An exact proof term for the current goal is (RltE_lt (apply_fun r11s x) (minus_SNo 1) Hylt).
L180432
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r11s x) den < mul_SNo (minus_SNo 1) den.
L180434
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r11s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L180435
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r11s x) den = apply_fun r11 x.
L180437
rewrite the current goal using HyEq (from left to right) at position 1.
L180437
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r11 x) den Hr11xS H23S H23ne0).
L180438
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L180440
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L180440
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L180442
We prove the intermediate claim Hr11x_lt_mden: apply_fun r11 x < minus_SNo den.
L180444
rewrite the current goal using HmulEq (from right to left).
L180444
rewrite the current goal using HrhsEq (from right to left).
L180445
An exact proof term for the current goal is HmulLt.
L180446
We prove the intermediate claim Hbad: Rlt (apply_fun r11 x) (minus_SNo den).
L180448
An exact proof term for the current goal is (RltI (apply_fun r11 x) (minus_SNo den) Hr11xR HmdenR Hr11x_lt_mden).
L180448
An exact proof term for the current goal is (Hnlt_lo Hbad).
L180449
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r11s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L180451
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r11s I Hr11s_contR HIcR Hr11s_I).
L180453
We prove the intermediate claim Hex_u12: ∃u12 : set, continuous_map X Tx I0 T0 u12 ∧ (∀x : set, x ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r11s ((closed_interval one_third 1) ∩ I) → apply_fun u12 x = one_third).
(*** further iteration scaffold: build u12 and g12 from residual r11s ***)
L180463
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r11s Hnorm HA Hr11s_cont).
L180463
Apply Hex_u12 to the current goal.
L180464
Let u12 be given.
L180465
Assume Hu12.
L180465
We prove the intermediate claim Hu12contI0: continuous_map X Tx I0 T0 u12.
L180467
We prove the intermediate claim Hu12AB: continuous_map X Tx I0 T0 u12 ∧ (∀x : set, x ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x = minus_SNo one_third).
L180471
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u12 ∧ (∀x : set, x ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r11s ((closed_interval one_third 1) ∩ I) → apply_fun u12 x = one_third) Hu12).
L180477
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u12) (∀x : set, x ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x = minus_SNo one_third) Hu12AB).
L180482
We prove the intermediate claim Hu12contR: continuous_map X Tx R R_standard_topology u12.
L180484
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L180485
We prove the intermediate claim HI0subR: I0 ⊆ R.
L180487
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L180487
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u12 Hu12contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L180495
Set den12 to be the term mul_SNo den11 den.
L180496
We prove the intermediate claim Hden12R: den12 ∈ R.
L180498
An exact proof term for the current goal is (real_mul_SNo den11 Hden11R den H23R).
L180498
We prove the intermediate claim Hden12pos: 0 < den12.
L180500
We prove the intermediate claim Hden11S: SNo den11.
L180501
An exact proof term for the current goal is (real_SNo den11 Hden11R).
L180501
An exact proof term for the current goal is (mul_SNo_pos_pos den11 den Hden11S H23S Hden11pos HdenPos).
L180502
Set u12s to be the term compose_fun X u12 (mul_const_fun den12).
L180503
We prove the intermediate claim Hu12s_cont: continuous_map X Tx R R_standard_topology u12s.
L180505
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u12 den12 HTx Hu12contR Hden12R Hden12pos).
L180505
Set g12 to be the term compose_fun X (pair_map X g11 u12s) add_fun_R.
L180506
We prove the intermediate claim Hg12cont: continuous_map X Tx R R_standard_topology g12.
L180508
An exact proof term for the current goal is (add_two_continuous_R X Tx g11 u12s HTx Hg11cont Hu12s_cont).
L180508
We prove the intermediate claim Hu12contA: continuous_map A Ta R R_standard_topology u12.
(*** further iteration scaffold: residual r12 on A and scaling r12s ***)
L180512
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u12 A HTx HAsubX Hu12contR).
L180512
Set u12neg to be the term compose_fun A u12 neg_fun.
L180513
We prove the intermediate claim Hu12neg_cont: continuous_map A Ta R R_standard_topology u12neg.
L180515
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u12 neg_fun Hu12contA Hnegcont).
L180516
We prove the intermediate claim Hr11s_contR: continuous_map A Ta R R_standard_topology r11s.
L180518
We prove the intermediate claim HTiEq''': Ti = subspace_topology R R_standard_topology I.
L180519
An exact proof term for the current goal is HTiEq.
L180519
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r11s Hr11s_cont HIcR R_standard_topology_is_topology_local HTiEq''').
L180521
Set r12 to be the term compose_fun A (pair_map A r11s u12neg) add_fun_R.
L180522
We prove the intermediate claim Hr12_cont: continuous_map A Ta R R_standard_topology r12.
L180524
An exact proof term for the current goal is (add_two_continuous_R A Ta r11s u12neg HTa Hr11s_contR Hu12neg_cont).
L180524
We prove the intermediate claim Hr12_apply: ∀x : set, x ∈ A → apply_fun r12 x = add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x)).
L180527
Let x be given.
L180527
Assume HxA: x ∈ A.
L180527
We prove the intermediate claim Hpimg: apply_fun (pair_map A r11s u12neg) x ∈ setprod R R.
L180529
rewrite the current goal using (pair_map_apply A R R r11s u12neg x HxA) (from left to right).
L180529
We prove the intermediate claim Hr11sxI: apply_fun r11s x ∈ I.
L180531
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r11s Hr11s_cont x HxA).
L180531
We prove the intermediate claim Hr11sxR: apply_fun r11s x ∈ R.
L180533
An exact proof term for the current goal is (HIcR (apply_fun r11s x) Hr11sxI).
L180533
We prove the intermediate claim Hu12negRx: apply_fun u12neg x ∈ R.
L180535
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u12neg Hu12neg_cont x HxA).
L180535
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r11s x) (apply_fun u12neg x) Hr11sxR Hu12negRx).
L180536
rewrite the current goal using (compose_fun_apply A (pair_map A r11s u12neg) add_fun_R x HxA) (from left to right).
L180537
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r11s u12neg) x) Hpimg) (from left to right) at position 1.
L180538
rewrite the current goal using (pair_map_apply A R R r11s u12neg x HxA) (from left to right).
L180539
rewrite the current goal using (tuple_2_0_eq (apply_fun r11s x) (apply_fun u12neg x)) (from left to right).
L180540
rewrite the current goal using (tuple_2_1_eq (apply_fun r11s x) (apply_fun u12neg x)) (from left to right).
L180541
rewrite the current goal using (compose_fun_apply A u12 neg_fun x HxA) (from left to right) at position 1.
L180542
We prove the intermediate claim Hu12Rx: apply_fun u12 x ∈ R.
L180544
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u12 Hu12contA x HxA).
L180544
rewrite the current goal using (neg_fun_apply (apply_fun u12 x) Hu12Rx) (from left to right) at position 1.
Use reflexivity.
L180546
We prove the intermediate claim Hr12_range: ∀x : set, x ∈ A → apply_fun r12 x ∈ I2.
L180548
We prove the intermediate claim Hu12AB: continuous_map X Tx I0 T0 u12 ∧ (∀x0 : set, x0 ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x0 = minus_SNo one_third).
L180552
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u12 ∧ (∀x0 : set, x0 ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r11s ((closed_interval one_third 1) ∩ I) → apply_fun u12 x0 = one_third) Hu12).
L180558
We prove the intermediate claim Hu12_on_B12: ∀x0 : set, x0 ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x0 = minus_SNo one_third.
L180562
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u12) (∀x0 : set, x0 ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x0 = minus_SNo one_third) Hu12AB).
L180566
We prove the intermediate claim Hu12_on_C12: ∀x0 : set, x0 ∈ preimage_of A r11s ((closed_interval one_third 1) ∩ I) → apply_fun u12 x0 = one_third.
L180570
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u12 ∧ (∀x0 : set, x0 ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r11s ((closed_interval one_third 1) ∩ I) → apply_fun u12 x0 = one_third) Hu12).
L180576
Let x be given.
L180577
Assume HxA: x ∈ A.
L180577
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L180578
Set I3 to be the term closed_interval one_third 1.
L180579
Set B12 to be the term preimage_of A r11s (I1 ∩ I).
L180580
Set C12 to be the term preimage_of A r11s (I3 ∩ I).
L180581
We prove the intermediate claim Hr11sIx: apply_fun r11s x ∈ I.
L180583
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r11s Hr11s_cont x HxA).
L180583
We prove the intermediate claim HB12_cases: x ∈ B12 ∨ ¬ (x ∈ B12).
L180585
An exact proof term for the current goal is (xm (x ∈ B12)).
L180585
Apply (HB12_cases (apply_fun r12 x ∈ I2)) to the current goal.
L180587
Assume HxB12: x ∈ B12.
L180587
We prove the intermediate claim Hu12eq: apply_fun u12 x = minus_SNo one_third.
L180589
An exact proof term for the current goal is (Hu12_on_B12 x HxB12).
L180589
We prove the intermediate claim Hr12eq: apply_fun r12 x = add_SNo (apply_fun r11s x) one_third.
L180591
rewrite the current goal using (Hr12_apply x HxA) (from left to right).
L180591
rewrite the current goal using Hu12eq (from left to right) at position 1.
L180592
We prove the intermediate claim H13R: one_third ∈ R.
L180594
An exact proof term for the current goal is one_third_in_R.
L180594
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L180596
rewrite the current goal using Hr12eq (from left to right).
L180597
We prove the intermediate claim Hr11sI1I: apply_fun r11s x ∈ I1 ∩ I.
L180599
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r11s x0 ∈ I1 ∩ I) x HxB12).
L180599
We prove the intermediate claim Hr11sI1: apply_fun r11s x ∈ I1.
L180601
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r11s x) Hr11sI1I).
L180601
We prove the intermediate claim H13R: one_third ∈ R.
L180603
An exact proof term for the current goal is one_third_in_R.
L180603
We prove the intermediate claim H23R: two_thirds ∈ R.
L180605
An exact proof term for the current goal is two_thirds_in_R.
L180605
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L180607
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L180607
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L180609
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L180609
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L180611
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L180611
We prove the intermediate claim Hr11sRx: apply_fun r11s x ∈ R.
L180613
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r11s x) Hr11sI1).
L180613
We prove the intermediate claim Hr11s_bounds: Rle (minus_SNo 1) (apply_fun r11s x) ∧ Rle (apply_fun r11s x) (minus_SNo one_third).
L180615
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r11s x) Hm1R Hm13R Hr11sI1).
L180616
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r11s x).
L180618
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r11s x)) (Rle (apply_fun r11s x) (minus_SNo one_third)) Hr11s_bounds).
L180619
We prove the intermediate claim HhiI1: Rle (apply_fun r11s x) (minus_SNo one_third).
L180621
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r11s x)) (Rle (apply_fun r11s x) (minus_SNo one_third)) Hr11s_bounds).
L180622
We prove the intermediate claim Hr12xR: add_SNo (apply_fun r11s x) one_third ∈ R.
L180624
An exact proof term for the current goal is (real_add_SNo (apply_fun r11s x) Hr11sRx one_third H13R).
L180624
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r11s x) one_third).
L180626
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r11s x) one_third Hm1R Hr11sRx H13R Hm1le).
L180626
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r11s x) one_third).
L180628
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L180628
An exact proof term for the current goal is Hlow_tmp.
L180629
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r11s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L180631
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r11s x) (minus_SNo one_third) one_third Hr11sRx Hm13R H13R HhiI1).
L180631
We prove the intermediate claim H13S: SNo one_third.
L180633
An exact proof term for the current goal is (real_SNo one_third H13R).
L180633
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r11s x) one_third) 0.
L180635
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L180635
An exact proof term for the current goal is Hup0_tmp.
L180636
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L180638
An exact proof term for the current goal is Rle_0_two_thirds.
L180638
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r11s x) one_third) two_thirds.
L180640
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r11s x) one_third) 0 two_thirds Hup0 H0le23).
L180640
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r11s x) one_third) Hm23R H23R Hr12xR Hlow Hup).
L180644
Assume HxnotB12: ¬ (x ∈ B12).
L180644
We prove the intermediate claim HC12_cases: x ∈ C12 ∨ ¬ (x ∈ C12).
L180646
An exact proof term for the current goal is (xm (x ∈ C12)).
L180646
Apply (HC12_cases (apply_fun r12 x ∈ I2)) to the current goal.
L180648
Assume HxC12: x ∈ C12.
L180648
We prove the intermediate claim Hu12eq: apply_fun u12 x = one_third.
L180650
An exact proof term for the current goal is (Hu12_on_C12 x HxC12).
L180650
We prove the intermediate claim Hr12eq: apply_fun r12 x = add_SNo (apply_fun r11s x) (minus_SNo one_third).
L180652
rewrite the current goal using (Hr12_apply x HxA) (from left to right).
L180652
rewrite the current goal using Hu12eq (from left to right) at position 1.
Use reflexivity.
L180654
rewrite the current goal using Hr12eq (from left to right).
L180655
We prove the intermediate claim Hr11sI3I: apply_fun r11s x ∈ I3 ∩ I.
L180657
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r11s x0 ∈ I3 ∩ I) x HxC12).
L180657
We prove the intermediate claim Hr11sI3: apply_fun r11s x ∈ I3.
L180659
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r11s x) Hr11sI3I).
L180659
We prove the intermediate claim H13R: one_third ∈ R.
L180661
An exact proof term for the current goal is one_third_in_R.
L180661
We prove the intermediate claim H23R: two_thirds ∈ R.
L180663
An exact proof term for the current goal is two_thirds_in_R.
L180663
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L180665
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L180665
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L180667
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L180667
We prove the intermediate claim Hr11sRx: apply_fun r11s x ∈ R.
L180669
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r11s x) Hr11sI3).
L180669
We prove the intermediate claim Hr11s_bounds: Rle one_third (apply_fun r11s x) ∧ Rle (apply_fun r11s x) 1.
L180671
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r11s x) H13R real_1 Hr11sI3).
L180671
We prove the intermediate claim HloI3: Rle one_third (apply_fun r11s x).
L180673
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r11s x)) (Rle (apply_fun r11s x) 1) Hr11s_bounds).
L180673
We prove the intermediate claim HhiI3: Rle (apply_fun r11s x) 1.
L180675
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r11s x)) (Rle (apply_fun r11s x) 1) Hr11s_bounds).
L180675
We prove the intermediate claim Hr12xR: add_SNo (apply_fun r11s x) (minus_SNo one_third) ∈ R.
L180677
An exact proof term for the current goal is (real_add_SNo (apply_fun r11s x) Hr11sRx (minus_SNo one_third) Hm13R).
L180677
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r11s x) (minus_SNo one_third)).
L180680
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r11s x) (minus_SNo one_third) H13R Hr11sRx Hm13R HloI3).
L180681
We prove the intermediate claim H13S: SNo one_third.
L180683
An exact proof term for the current goal is (real_SNo one_third H13R).
L180683
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r11s x) (minus_SNo one_third)).
L180685
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L180685
An exact proof term for the current goal is H0le_tmp.
L180686
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L180688
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L180688
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r11s x) (minus_SNo one_third)).
L180690
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r11s x) (minus_SNo one_third)) Hm23le0 H0le).
L180692
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r11s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L180695
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r11s x) 1 (minus_SNo one_third) Hr11sRx real_1 Hm13R HhiI3).
L180696
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L180698
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L180698
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L180699
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r11s x) (minus_SNo one_third)) two_thirds.
L180701
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r11s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L180704
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r11s x) (minus_SNo one_third)) Hm23R H23R Hr12xR Hlow Hup).
L180708
Assume HxnotC12: ¬ (x ∈ C12).
L180708
We prove the intermediate claim HxX: x ∈ X.
L180710
An exact proof term for the current goal is (HAsubX x HxA).
L180710
We prove the intermediate claim HnotI1: ¬ (apply_fun r11s x ∈ I1).
L180712
Assume Hr11sI1': apply_fun r11s x ∈ I1.
L180712
We prove the intermediate claim Hr11sI1I: apply_fun r11s x ∈ I1 ∩ I.
L180714
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r11s x) Hr11sI1' Hr11sIx).
L180714
We prove the intermediate claim HxB12': x ∈ B12.
L180716
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r11s x0 ∈ I1 ∩ I) x HxA Hr11sI1I).
L180716
Apply FalseE to the current goal.
L180717
An exact proof term for the current goal is (HxnotB12 HxB12').
L180718
We prove the intermediate claim HnotI3: ¬ (apply_fun r11s x ∈ I3).
L180720
Assume Hr11sI3': apply_fun r11s x ∈ I3.
L180720
We prove the intermediate claim Hr11sI3I: apply_fun r11s x ∈ I3 ∩ I.
L180722
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r11s x) Hr11sI3' Hr11sIx).
L180722
We prove the intermediate claim HxC12': x ∈ C12.
L180724
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r11s x0 ∈ I3 ∩ I) x HxA Hr11sI3I).
L180724
Apply FalseE to the current goal.
L180725
An exact proof term for the current goal is (HxnotC12 HxC12').
L180726
We prove the intermediate claim Hr11sRx: apply_fun r11s x ∈ R.
L180728
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r11s x) Hr11sIx).
L180728
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L180730
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L180730
We prove the intermediate claim H13R: one_third ∈ R.
L180732
An exact proof term for the current goal is one_third_in_R.
L180732
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L180734
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L180734
We prove the intermediate claim Hr11s_boundsI: Rle (minus_SNo 1) (apply_fun r11s x) ∧ Rle (apply_fun r11s x) 1.
L180736
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r11s x) Hm1R real_1 Hr11sIx).
L180736
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r11s x) (minus_SNo 1)).
L180738
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r11s x) (andEL (Rle (minus_SNo 1) (apply_fun r11s x)) (Rle (apply_fun r11s x) 1) Hr11s_boundsI)).
L180739
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r11s x)).
L180741
An exact proof term for the current goal is (RleE_nlt (apply_fun r11s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r11s x)) (Rle (apply_fun r11s x) 1) Hr11s_boundsI)).
L180742
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r11s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r11s x).
L180744
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r11s x) Hm1R Hm13R Hr11sRx HnotI1).
L180745
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r11s x).
L180747
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r11s x))) to the current goal.
L180748
Assume Hbad: Rlt (apply_fun r11s x) (minus_SNo 1).
L180748
Apply FalseE to the current goal.
L180749
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L180751
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r11s x).
L180751
An exact proof term for the current goal is Hok.
L180752
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r11s x) (minus_SNo one_third)).
L180754
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r11s x) Hm13lt_fx).
L180754
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r11s x) one_third ∨ Rlt 1 (apply_fun r11s x).
L180756
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r11s x) H13R real_1 Hr11sRx HnotI3).
L180757
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r11s x) one_third.
L180759
Apply (HnotI3_cases (Rlt (apply_fun r11s x) one_third)) to the current goal.
L180760
Assume Hok: Rlt (apply_fun r11s x) one_third.
L180760
An exact proof term for the current goal is Hok.
L180762
Assume Hbad: Rlt 1 (apply_fun r11s x).
L180762
Apply FalseE to the current goal.
L180763
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L180764
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r11s x)).
L180766
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r11s x) one_third Hfx_lt_13).
L180766
We prove the intermediate claim Hr11sI0: apply_fun r11s x ∈ I0.
L180768
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L180769
We prove the intermediate claim HxSep: apply_fun r11s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L180771
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r11s x) Hr11sRx (andI (¬ (Rlt (apply_fun r11s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r11s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L180775
rewrite the current goal using HI0_def (from left to right).
L180776
An exact proof term for the current goal is HxSep.
L180777
We prove the intermediate claim Hu12funI0: function_on u12 X I0.
L180779
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u12 Hu12contI0).
L180779
We prove the intermediate claim Hu12xI0: apply_fun u12 x ∈ I0.
L180781
An exact proof term for the current goal is (Hu12funI0 x HxX).
L180781
We prove the intermediate claim Hu12xR: apply_fun u12 x ∈ R.
L180783
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u12 x) Hu12xI0).
L180783
We prove the intermediate claim Hm_u12x_R: minus_SNo (apply_fun u12 x) ∈ R.
L180785
An exact proof term for the current goal is (real_minus_SNo (apply_fun u12 x) Hu12xR).
L180785
rewrite the current goal using (Hr12_apply x HxA) (from left to right).
L180786
We prove the intermediate claim Hr12xR: add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x)) ∈ R.
L180788
An exact proof term for the current goal is (real_add_SNo (apply_fun r11s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r11s x) Hr11sI0) (minus_SNo (apply_fun u12 x)) Hm_u12x_R).
L180790
We prove the intermediate claim H23R: two_thirds ∈ R.
L180792
An exact proof term for the current goal is two_thirds_in_R.
L180792
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L180794
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L180794
We prove the intermediate claim Hr11s_bounds0: Rle (minus_SNo one_third) (apply_fun r11s x) ∧ Rle (apply_fun r11s x) one_third.
L180796
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r11s x) Hm13R H13R Hr11sI0).
L180796
We prove the intermediate claim Hu12_bounds0: Rle (minus_SNo one_third) (apply_fun u12 x) ∧ Rle (apply_fun u12 x) one_third.
L180798
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u12 x) Hm13R H13R Hu12xI0).
L180798
We prove the intermediate claim Hr11s_le_13: Rle (apply_fun r11s x) one_third.
L180800
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r11s x)) (Rle (apply_fun r11s x) one_third) Hr11s_bounds0).
L180803
We prove the intermediate claim Hr11s_ge_m13: Rle (minus_SNo one_third) (apply_fun r11s x).
L180805
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r11s x)) (Rle (apply_fun r11s x) one_third) Hr11s_bounds0).
L180808
We prove the intermediate claim Hu12_le_13: Rle (apply_fun u12 x) one_third.
L180810
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u12 x)) (Rle (apply_fun u12 x) one_third) Hu12_bounds0).
L180813
We prove the intermediate claim Hu12_ge_m13: Rle (minus_SNo one_third) (apply_fun u12 x).
L180815
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u12 x)) (Rle (apply_fun u12 x) one_third) Hu12_bounds0).
L180818
We prove the intermediate claim Hm13_le_mu12: Rle (minus_SNo one_third) (minus_SNo (apply_fun u12 x)).
L180820
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u12 x) one_third Hu12_le_13).
L180820
We prove the intermediate claim Hmu12_le_13: Rle (minus_SNo (apply_fun u12 x)) one_third.
L180822
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u12 x)) (minus_SNo (minus_SNo one_third)).
L180823
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u12 x) Hu12_ge_m13).
L180823
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L180824
An exact proof term for the current goal is Htmp.
L180825
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u12 x))).
L180828
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u12 x)) Hm13R Hm13R Hm_u12x_R Hm13_le_mu12).
L180829
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u12 x))) (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))).
L180832
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r11s x) (minus_SNo (apply_fun u12 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r11s x) Hr11sI0) Hm_u12x_R Hr11s_ge_m13).
L180835
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))).
L180838
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u12 x))) (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))) Hlow1 Hlow2).
L180841
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))).
L180843
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L180843
An exact proof term for the current goal is Hlow_tmp.
L180844
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))) (add_SNo (apply_fun r11s x) one_third).
L180847
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r11s x) (minus_SNo (apply_fun u12 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r11s x) Hr11sI0) Hm_u12x_R H13R Hmu12_le_13).
L180849
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r11s x) one_third) (add_SNo one_third one_third).
L180852
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r11s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r11s x) Hr11sI0) H13R H13R Hr11s_le_13).
L180854
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))) (add_SNo one_third one_third).
L180857
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))) (add_SNo (apply_fun r11s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L180860
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L180862
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))) two_thirds.
L180864
rewrite the current goal using Hdef23 (from left to right) at position 1.
L180864
An exact proof term for the current goal is Hup_tmp.
L180865
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))) Hm23R H23R Hr12xR Hlow Hup).
L180868
Set r12s to be the term compose_fun A r12 (div_const_fun den).
L180869
We prove the intermediate claim Hr12s_cont: continuous_map A Ta I Ti r12s.
L180871
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
(*** structured: prove continuity into R, then restrict range to I ***)
L180873
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L180873
We prove the intermediate claim Hr12s_contR: continuous_map A Ta R R_standard_topology r12s.
L180875
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r12 (div_const_fun den) Hr12_cont Hdivcont).
L180876
We prove the intermediate claim Hr12s_I: ∀x : set, x ∈ A → apply_fun r12s x ∈ I.
L180878
Let x be given.
L180878
Assume HxA: x ∈ A.
L180878
We prove the intermediate claim Hr12xI2: apply_fun r12 x ∈ I2.
L180880
An exact proof term for the current goal is (Hr12_range x HxA).
L180880
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L180882
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L180882
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L180884
An exact proof term for the current goal is (real_minus_SNo den H23R).
L180884
We prove the intermediate claim Hr12xR: apply_fun r12 x ∈ R.
L180886
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r12 x) Hr12xI2).
L180886
We prove the intermediate claim Hr12xS: SNo (apply_fun r12 x).
L180888
An exact proof term for the current goal is (real_SNo (apply_fun r12 x) Hr12xR).
L180888
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r12 x) ∧ Rle (apply_fun r12 x) den.
L180890
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r12 x) HmdenR H23R Hr12xI2).
L180890
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r12 x).
L180892
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r12 x)) (Rle (apply_fun r12 x) den) Hbounds).
L180894
We prove the intermediate claim Hhi: Rle (apply_fun r12 x) den.
L180896
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r12 x)) (Rle (apply_fun r12 x) den) Hbounds).
L180898
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r12 x)).
L180900
An exact proof term for the current goal is (RleE_nlt (apply_fun r12 x) den Hhi).
L180900
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r12 x) (minus_SNo den)).
L180902
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r12 x) Hlo).
L180902
We prove the intermediate claim HyEq: apply_fun r12s x = div_SNo (apply_fun r12 x) den.
L180904
rewrite the current goal using (compose_fun_apply A r12 (div_const_fun den) x HxA) (from left to right).
L180904
rewrite the current goal using (div_const_fun_apply den (apply_fun r12 x) H23R Hr12xR) (from left to right).
Use reflexivity.
L180906
We prove the intermediate claim HyR: apply_fun r12s x ∈ R.
L180908
rewrite the current goal using HyEq (from left to right).
L180908
An exact proof term for the current goal is (real_div_SNo (apply_fun r12 x) Hr12xR den H23R).
L180909
We prove the intermediate claim HyS: SNo (apply_fun r12s x).
L180911
An exact proof term for the current goal is (real_SNo (apply_fun r12s x) HyR).
L180911
We prove the intermediate claim Hy_le_1: Rle (apply_fun r12s x) 1.
L180913
Apply (RleI (apply_fun r12s x) 1 HyR real_1) to the current goal.
L180913
We will prove ¬ (Rlt 1 (apply_fun r12s x)).
L180914
Assume H1lt: Rlt 1 (apply_fun r12s x).
L180915
We prove the intermediate claim H1lty: 1 < apply_fun r12s x.
L180917
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r12s x) H1lt).
L180917
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r12s x) den.
L180919
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r12s x) den SNo_1 HyS H23S H23pos H1lty).
L180919
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r12s x) den.
L180921
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L180921
An exact proof term for the current goal is HmulLt.
L180922
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r12s x) den = apply_fun r12 x.
L180924
rewrite the current goal using HyEq (from left to right) at position 1.
L180924
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r12 x) den Hr12xS H23S H23ne0).
L180925
We prove the intermediate claim Hden_lt_r12x: den < apply_fun r12 x.
L180927
rewrite the current goal using HmulEq (from right to left).
L180927
An exact proof term for the current goal is HmulLt'.
L180928
We prove the intermediate claim Hbad: Rlt den (apply_fun r12 x).
L180930
An exact proof term for the current goal is (RltI den (apply_fun r12 x) H23R Hr12xR Hden_lt_r12x).
L180930
An exact proof term for the current goal is (Hnlt_hi Hbad).
L180931
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r12s x).
L180933
Apply (RleI (minus_SNo 1) (apply_fun r12s x) Hm1R HyR) to the current goal.
L180933
We will prove ¬ (Rlt (apply_fun r12s x) (minus_SNo 1)).
L180934
Assume Hylt: Rlt (apply_fun r12s x) (minus_SNo 1).
L180935
We prove the intermediate claim Hylts: apply_fun r12s x < minus_SNo 1.
L180937
An exact proof term for the current goal is (RltE_lt (apply_fun r12s x) (minus_SNo 1) Hylt).
L180937
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r12s x) den < mul_SNo (minus_SNo 1) den.
L180939
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r12s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L180940
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r12s x) den = apply_fun r12 x.
L180942
rewrite the current goal using HyEq (from left to right) at position 1.
L180942
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r12 x) den Hr12xS H23S H23ne0).
L180943
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L180945
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L180945
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L180947
We prove the intermediate claim Hr12x_lt_mden: apply_fun r12 x < minus_SNo den.
L180949
rewrite the current goal using HmulEq (from right to left).
L180949
rewrite the current goal using HrhsEq (from right to left).
L180950
An exact proof term for the current goal is HmulLt.
L180951
We prove the intermediate claim Hbad: Rlt (apply_fun r12 x) (minus_SNo den).
L180953
An exact proof term for the current goal is (RltI (apply_fun r12 x) (minus_SNo den) Hr12xR HmdenR Hr12x_lt_mden).
L180953
An exact proof term for the current goal is (Hnlt_lo Hbad).
L180954
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r12s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L180956
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r12s I Hr12s_contR HIcR Hr12s_I).
L180958
We prove the intermediate claim Hexfn: ∃fn : set, function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n)) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I) ∧ (∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x)) ∧ uniform_cauchy_metric X R R_bounded_metric fn.
(*** pending: continue the iteration further and prove uniform convergence ***)
L180970
Set I0 to be the term closed_interval (minus_SNo one_third) one_third.
(*** TeX Step II (in progress): define a full correction sequence by nat_primrec and prove it has the needed properties. ***)
L180972
Set T0 to be the term closed_interval_topology (minus_SNo one_third) one_third.
(*** We keep the already-built finite scaffold above, but the infinite sequence is defined independently. ***)
L180973
Set u_of to be the term (λr : set ⇒ Eps_i (λu : set ⇒ continuous_map X Tx I0 T0 u ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun u x = one_third))).
L180980
We prove the intermediate claim Hu_of_prop: ∀r : set, continuous_map A Ta I Ti r → continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x = one_third).
(*** helper: Step I witness extracted via Eps_i ***)
L180988
Let r be given.
L180988
Assume Hr: continuous_map A Ta I Ti r.
L180988
Set P to be the term (λu : set ⇒ continuous_map X Tx I0 T0 u ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun u x = one_third)).
L180994
We prove the intermediate claim Hexu: ∃u : set, P u.
L180996
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r Hnorm HA Hr).
L180996
Apply Hexu to the current goal.
L180997
Let u be given.
L180998
Assume Hu.
L180998
We prove the intermediate claim HP: P (Eps_i P).
L181000
An exact proof term for the current goal is (Eps_i_ax P u Hu).
L181000
We prove the intermediate claim Hu_eq: u_of r = Eps_i P.
Use reflexivity.
L181002
rewrite the current goal using Hu_eq (from left to right).
L181003
An exact proof term for the current goal is HP.
L181004
Set g0g to be the term graph X (λx : set ⇒ apply_fun g0 x).
L181006
Set BaseState to be the term (g0g,(f1s,den)).
(*** base stage: graphify g0 so that it lies in function_space X R ***)
L181007
Set StepState to be the term (λn st : set ⇒ (compose_fun X (pair_map X (st 0) (compose_fun X (u_of ((st 1) 0)) (mul_const_fun ((st 1) 1)))) add_fun_R,(compose_fun A (compose_fun A (pair_map A ((st 1) 0) (compose_fun A (u_of ((st 1) 0)) neg_fun)) add_fun_R) (div_const_fun den),mul_SNo ((st 1) 1) den))).
L181023
Set State to be the term (λn : set ⇒ nat_primrec BaseState StepState n).
L181024
Set fn to be the term graph ω (λn : set ⇒ (State n) 0).
L181025
We use fn to witness the existential quantifier.
L181026
We prove the intermediate claim HInv_cpos: ∀n : set, n ∈ ω → ((State n) 1) 1 ∈ R ∧ 0 < ((State n) 1) 1.
(*** helper: scalar component stays positive ***)
L181029
We prove the intermediate claim HInv_nat: ∀n : set, nat_p n → ((State n) 1) 1 ∈ R ∧ 0 < ((State n) 1) 1.
L181030
Apply nat_ind to the current goal.
L181031
We will prove ((State 0) 1) 1 ∈ R ∧ 0 < ((State 0) 1) 1.
L181031
We prove the intermediate claim H0: State 0 = BaseState.
L181033
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L181033
rewrite the current goal using H0 (from left to right).
L181034
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L181036
rewrite the current goal using HBase (from left to right).
L181037
We prove the intermediate claim HtEq: ((g0g,(f1s,den)) 1) 1 = den.
L181039
rewrite the current goal using (tuple_2_1_eq g0g (f1s,den)) (from left to right) at position 1.
L181039
rewrite the current goal using (tuple_2_1_eq f1s den) (from left to right) at position 1.
Use reflexivity.
L181041
Apply andI to the current goal.
L181043
rewrite the current goal using HtEq (from left to right).
L181043
An exact proof term for the current goal is HdenR.
L181045
rewrite the current goal using HtEq (from left to right).
L181045
An exact proof term for the current goal is HdenPos.
L181047
Let n be given.
L181047
Assume HnNat: nat_p n.
L181047
Assume IH: ((State n) 1) 1 ∈ R ∧ 0 < ((State n) 1) 1.
L181048
We will prove ((State (ordsucc n)) 1) 1 ∈ R ∧ 0 < ((State (ordsucc n)) 1) 1.
L181049
We prove the intermediate claim HS: State (ordsucc n) = StepState n (State n).
L181051
An exact proof term for the current goal is (nat_primrec_S BaseState StepState n HnNat).
L181051
rewrite the current goal using HS (from left to right).
L181052
Set st to be the term State n.
L181053
Set g to be the term st 0.
L181054
Set rpack to be the term st 1.
L181055
Set r to be the term rpack 0.
L181056
Set c to be the term rpack 1.
L181057
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L181060
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L181065
Set cNew to be the term mul_SNo c den.
L181066
We prove the intermediate claim Hdef: StepState n st = (gNew,(rNew,cNew)).
Use reflexivity.
L181068
We prove the intermediate claim HtEq: ((StepState n st) 1) 1 = cNew.
L181070
We prove the intermediate claim Hinner: (StepState n st) 1 = (rNew,cNew).
L181071
rewrite the current goal using Hdef (from left to right).
L181071
An exact proof term for the current goal is (tuple_2_1_eq gNew (rNew,cNew)).
L181072
rewrite the current goal using Hinner (from left to right).
L181073
An exact proof term for the current goal is (tuple_2_1_eq rNew cNew).
L181074
We prove the intermediate claim Hc_eq: c = ((State n) 1) 1.
Use reflexivity.
L181076
We prove the intermediate claim HcR0: ((State n) 1) 1 ∈ R.
L181078
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) IH).
L181078
We prove the intermediate claim Hcpos0: 0 < ((State n) 1) 1.
L181080
An exact proof term for the current goal is (andER (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) IH).
L181080
We prove the intermediate claim HcR: c ∈ R.
L181082
rewrite the current goal using Hc_eq (from left to right).
L181082
An exact proof term for the current goal is HcR0.
L181082
We prove the intermediate claim Hcpos: 0 < c.
L181084
rewrite the current goal using Hc_eq (from left to right).
L181084
An exact proof term for the current goal is Hcpos0.
L181084
We prove the intermediate claim HcS: SNo c.
L181086
An exact proof term for the current goal is (real_SNo c HcR).
L181086
We prove the intermediate claim HdenS: SNo den.
L181088
An exact proof term for the current goal is (real_SNo den HdenR).
L181088
We prove the intermediate claim HcNewR: cNew ∈ R.
L181090
An exact proof term for the current goal is (real_mul_SNo c HcR den HdenR).
L181090
We prove the intermediate claim HcNewPos: 0 < cNew.
L181092
An exact proof term for the current goal is (mul_SNo_pos_pos c den HcS HdenS Hcpos HdenPos).
L181092
Apply andI to the current goal.
L181094
rewrite the current goal using HtEq (from left to right).
L181094
An exact proof term for the current goal is HcNewR.
L181096
rewrite the current goal using HtEq (from left to right).
L181096
An exact proof term for the current goal is HcNewPos.
L181097
Let n be given.
L181098
Assume HnO: n ∈ ω.
L181098
An exact proof term for the current goal is (HInv_nat n (omega_nat_p n HnO)).
L181099
We prove the intermediate claim HInv_c_lt1: ∀n : set, n ∈ ω → ((State n) 1) 1 < 1.
(*** helper: scalar component is always strictly below 1 ***)
L181102
We prove the intermediate claim HInv_nat: ∀n : set, nat_p n → ((State n) 1) 1 < 1.
L181103
Apply nat_ind to the current goal.
L181104
We will prove ((State 0) 1) 1 < 1.
L181104
We prove the intermediate claim H0: State 0 = BaseState.
L181106
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L181106
rewrite the current goal using H0 (from left to right).
L181107
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L181109
rewrite the current goal using HBase (from left to right).
L181110
We prove the intermediate claim HtEq: ((g0g,(f1s,den)) 1) 1 = den.
L181112
rewrite the current goal using (tuple_2_1_eq g0g (f1s,den)) (from left to right) at position 1.
L181112
rewrite the current goal using (tuple_2_1_eq f1s den) (from left to right) at position 1.
Use reflexivity.
L181114
rewrite the current goal using HtEq (from left to right).
L181115
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
L181117
rewrite the current goal using HdenDef (from left to right).
L181118
An exact proof term for the current goal is (RltE_lt two_thirds 1 Rlt_two_thirds_1).
L181120
Let n be given.
L181120
Assume HnNat: nat_p n.
L181120
Assume IH: ((State n) 1) 1 < 1.
L181121
We will prove ((State (ordsucc n)) 1) 1 < 1.
L181122
We prove the intermediate claim HS: State (ordsucc n) = StepState n (State n).
L181124
An exact proof term for the current goal is (nat_primrec_S BaseState StepState n HnNat).
L181124
rewrite the current goal using HS (from left to right).
L181125
Set st to be the term State n.
L181126
Set g to be the term st 0.
L181127
Set rpack to be the term st 1.
L181128
Set r to be the term rpack 0.
L181129
Set c to be the term rpack 1.
L181130
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L181133
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L181138
Set cNew to be the term mul_SNo c den.
L181139
We prove the intermediate claim Hdef: StepState n st = (gNew,(rNew,cNew)).
Use reflexivity.
L181141
We prove the intermediate claim HtEq: ((StepState n st) 1) 1 = cNew.
L181143
We prove the intermediate claim Hinner: (StepState n st) 1 = (rNew,cNew).
L181144
rewrite the current goal using Hdef (from left to right).
L181144
An exact proof term for the current goal is (tuple_2_1_eq gNew (rNew,cNew)).
L181145
rewrite the current goal using Hinner (from left to right).
L181146
An exact proof term for the current goal is (tuple_2_1_eq rNew cNew).
L181147
rewrite the current goal using HtEq (from left to right).
L181148
We prove the intermediate claim HnO: n ∈ ω.
L181150
An exact proof term for the current goal is (nat_p_omega n HnNat).
L181150
We prove the intermediate claim Hcpack: ((State n) 1) 1 ∈ R ∧ 0 < ((State n) 1) 1.
L181152
An exact proof term for the current goal is (HInv_cpos n HnO).
L181152
We prove the intermediate claim HcR: ((State n) 1) 1 ∈ R.
L181154
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) Hcpack).
L181154
We prove the intermediate claim Hcpos: 0 < ((State n) 1) 1.
L181156
An exact proof term for the current goal is (andER (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) Hcpack).
L181156
We prove the intermediate claim HcS: SNo (((State n) 1) 1).
L181158
An exact proof term for the current goal is (real_SNo (((State n) 1) 1) HcR).
L181158
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
L181160
We prove the intermediate claim HdenLt1: den < 1.
L181162
rewrite the current goal using HdenDef (from left to right).
L181162
An exact proof term for the current goal is (RltE_lt two_thirds 1 Rlt_two_thirds_1).
L181163
We prove the intermediate claim HdenS: SNo den.
L181165
An exact proof term for the current goal is (real_SNo den HdenR).
L181165
We prove the intermediate claim HmulLt: mul_SNo den (((State n) 1) 1) < ((State n) 1) 1.
L181167
An exact proof term for the current goal is (mul_SNo_Lt1_pos_Lt den (((State n) 1) 1) HdenS HcS HdenLt1 Hcpos).
L181167
We prove the intermediate claim HmulEq: mul_SNo den (((State n) 1) 1) = mul_SNo (((State n) 1) 1) den.
L181169
An exact proof term for the current goal is (mul_SNo_com den (((State n) 1) 1) HdenS HcS).
L181169
We prove the intermediate claim HcNewLt: cNew < ((State n) 1) 1.
L181171
We prove the intermediate claim HcEq: c = ((State n) 1) 1.
Use reflexivity.
L181172
rewrite the current goal using HcEq (from left to right) at position 1.
L181173
rewrite the current goal using HmulEq (from right to left) at position 1.
L181174
An exact proof term for the current goal is HmulLt.
L181175
We prove the intermediate claim HcNewR: cNew ∈ R.
L181177
We prove the intermediate claim HcEq: c = ((State n) 1) 1.
Use reflexivity.
L181178
rewrite the current goal using HcEq (from left to right).
L181179
An exact proof term for the current goal is (real_mul_SNo (((State n) 1) 1) HcR den HdenR).
L181180
We prove the intermediate claim HcNewS: SNo cNew.
L181182
An exact proof term for the current goal is (real_SNo cNew HcNewR).
L181182
An exact proof term for the current goal is (SNoLt_tra cNew (((State n) 1) 1) 1 HcNewS HcS SNo_1 HcNewLt IH).
L181188
Let n be given.
L181189
Assume HnO: n ∈ ω.
L181189
An exact proof term for the current goal is (HInv_nat n (omega_nat_p n HnO)).
L181190
We prove the intermediate claim HInv_r_contI: ∀n : set, n ∈ ω → continuous_map A Ta I Ti (((State n) 1) 0).
(*** helper: residual component stays a continuous I-valued map on A ***)
L181193
We prove the intermediate claim HInv_nat: ∀n : set, nat_p n → continuous_map A Ta I Ti (((State n) 1) 0).
L181194
Apply nat_ind to the current goal.
L181195
We will prove continuous_map A Ta I Ti (((State 0) 1) 0).
L181195
We prove the intermediate claim H0: State 0 = BaseState.
L181197
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L181197
rewrite the current goal using H0 (from left to right).
L181198
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L181200
rewrite the current goal using HBase (from left to right).
L181201
We prove the intermediate claim Hr0eq: (((g0g,(f1s,den)) 1) 0) = f1s.
L181203
rewrite the current goal using (tuple_2_1_eq g0g (f1s,den)) (from left to right) at position 1.
L181203
rewrite the current goal using (tuple_2_0_eq f1s den) (from left to right) at position 1.
Use reflexivity.
L181205
rewrite the current goal using Hr0eq (from left to right).
L181206
An exact proof term for the current goal is Hf1s_cont.
L181208
Let n be given.
L181208
Assume HnNat: nat_p n.
L181208
Assume IH: continuous_map A Ta I Ti (((State n) 1) 0).
L181209
We will prove continuous_map A Ta I Ti (((State (ordsucc n)) 1) 0).
L181210
We prove the intermediate claim HS: State (ordsucc n) = StepState n (State n).
L181212
An exact proof term for the current goal is (nat_primrec_S BaseState StepState n HnNat).
L181212
rewrite the current goal using HS (from left to right).
L181213
Set st to be the term State n.
L181214
Set g to be the term st 0.
L181215
Set rpack to be the term st 1.
L181216
Set r to be the term rpack 0.
L181217
Set c to be the term rpack 1.
L181218
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L181221
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L181226
Set cNew to be the term mul_SNo c den.
L181227
We prove the intermediate claim Hdef: StepState n st = (gNew,(rNew,cNew)).
Use reflexivity.
L181229
We prove the intermediate claim HrEq: ((StepState n st) 1) 0 = rNew.
L181231
We prove the intermediate claim Hinner: (StepState n st) 1 = (rNew,cNew).
L181232
rewrite the current goal using Hdef (from left to right).
L181232
An exact proof term for the current goal is (tuple_2_1_eq gNew (rNew,cNew)).
L181233
rewrite the current goal using Hinner (from left to right).
L181234
An exact proof term for the current goal is (tuple_2_0_eq rNew cNew).
L181235
rewrite the current goal using HrEq (from left to right).
L181236
We prove the intermediate claim HIcR: I ⊆ R.
(*** continuity of rNew into I, via range restriction ***)
L181239
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1).
L181239
We prove the intermediate claim HTiEq: Ti = subspace_topology R R_standard_topology I.
Use reflexivity.
L181241
We prove the intermediate claim Hr_contI: continuous_map A Ta I Ti r.
L181243
We prove the intermediate claim HrEq0: r = ((State n) 1) 0.
Use reflexivity.
L181244
rewrite the current goal using HrEq0 (from left to right).
L181245
An exact proof term for the current goal is IH.
L181246
We prove the intermediate claim Hr_contR: continuous_map A Ta R R_standard_topology r.
L181248
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r Hr_contI HIcR R_standard_topology_is_topology_local HTiEq).
L181249
We prove the intermediate claim Hu_pack: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x = one_third).
L181256
An exact proof term for the current goal is (Hu_of_prop r Hr_contI).
L181256
We prove the intermediate claim Hu_left: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third).
L181261
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x = one_third) Hu_pack).
L181267
We prove the intermediate claim Hu_contI0: continuous_map X Tx I0 T0 (u_of r).
L181269
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r)) (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third) Hu_left).
L181273
We prove the intermediate claim HI0cR: I0 ⊆ R.
L181275
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L181275
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L181277
We prove the intermediate claim Hu_contR: continuous_map X Tx R R_standard_topology (u_of r).
L181279
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology (u_of r) Hu_contI0 HI0cR R_standard_topology_is_topology_local HT0eq).
L181280
We prove the intermediate claim HTx: topology_on X Tx.
L181282
An exact proof term for the current goal is (normal_space_topology_on X Tx Hnorm).
L181282
We prove the intermediate claim HAsubX: A ⊆ X.
L181284
An exact proof term for the current goal is (closed_in_subset X Tx A HA).
L181284
We prove the intermediate claim HuA_contR: continuous_map A Ta R R_standard_topology (u_of r).
L181286
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology (u_of r) A HTx HAsubX Hu_contR).
L181286
We prove the intermediate claim Hneg_cont: continuous_map R R_standard_topology R R_standard_topology neg_fun.
L181288
An exact proof term for the current goal is neg_fun_continuous.
L181288
We prove the intermediate claim HnegA_contR: continuous_map A Ta R R_standard_topology (compose_fun A (u_of r) neg_fun).
L181290
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology (u_of r) neg_fun HuA_contR Hneg_cont).
L181291
We prove the intermediate claim Hpair_cont: continuous_map A Ta (setprod R R) (product_topology R R_standard_topology R R_standard_topology) (pair_map A r (compose_fun A (u_of r) neg_fun)).
L181295
An exact proof term for the current goal is (maps_into_products_axiom A Ta R R_standard_topology R R_standard_topology r (compose_fun A (u_of r) neg_fun) Hr_contR HnegA_contR).
L181296
We prove the intermediate claim Hadd_cont: continuous_map (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology add_fun_R.
L181300
An exact proof term for the current goal is add_fun_R_continuous.
L181300
We prove the intermediate claim Hh_cont: continuous_map A Ta R R_standard_topology (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R).
L181303
An exact proof term for the current goal is (composition_continuous A Ta (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R Hpair_cont Hadd_cont).
L181307
We prove the intermediate claim Hdiv_cont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L181309
An exact proof term for the current goal is (div_const_fun_continuous_pos den HdenR HdenPos).
L181309
We prove the intermediate claim HrNew_contR: continuous_map A Ta R R_standard_topology rNew.
L181311
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den) Hh_cont Hdiv_cont).
L181314
We prove the intermediate claim HrNew_img: ∀x : set, x ∈ A → apply_fun rNew x ∈ I.
L181316
Let x be given.
L181316
Assume HxA: x ∈ A.
L181316
Set rNum to be the term compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R.
L181317
Set I2 to be the term closed_interval (minus_SNo den) den.
L181318
We prove the intermediate claim HrNumI2: apply_fun rNum x ∈ I2.
L181320
We prove the intermediate claim HxX: x ∈ X.
(*** proof scaffold: rNum x = r x - u_of r x, then show bounds in [-den,den] ***)
L181322
An exact proof term for the current goal is (HAsubX x HxA).
L181322
We prove the intermediate claim Hfun_r: function_on r A I.
L181324
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r Hr_contI).
L181324
We prove the intermediate claim HrxI: apply_fun r x ∈ I.
L181326
An exact proof term for the current goal is (Hfun_r x HxA).
L181326
We prove the intermediate claim HrxR: apply_fun r x ∈ R.
L181328
An exact proof term for the current goal is (HIcR (apply_fun r x) HrxI).
L181328
We prove the intermediate claim Hfun_u: function_on (u_of r) X I0.
L181330
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 (u_of r) Hu_contI0).
L181330
We prove the intermediate claim HuxI0: apply_fun (u_of r) x ∈ I0.
L181332
An exact proof term for the current goal is (Hfun_u x HxX).
L181332
We prove the intermediate claim HuxR: apply_fun (u_of r) x ∈ R.
L181334
An exact proof term for the current goal is (HI0cR (apply_fun (u_of r) x) HuxI0).
L181334
We prove the intermediate claim HrNumEq: apply_fun rNum x = add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x)).
L181337
An exact proof term for the current goal is (add_of_pair_map_neg_apply A r (u_of r) x HxA HrxR HuxR).
L181337
Apply (xm (Rlt (apply_fun r x) (minus_SNo one_third))) to the current goal.
L181339
Assume Hr_lt_left: Rlt (apply_fun r x) (minus_SNo one_third).
L181339
We prove the intermediate claim Hu_left_eq: ∀y : set, y ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) y = minus_SNo one_third.
L181343
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 (u_of r)) (∀y0 : set, y0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) y0 = minus_SNo one_third) Hu_left).
L181346
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L181348
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L181348
We prove the intermediate claim Hm13R: (minus_SNo one_third) ∈ R.
L181350
An exact proof term for the current goal is (real_minus_SNo one_third one_third_in_R).
L181350
We prove the intermediate claim HrxB: Rle (minus_SNo 1) (apply_fun r x) ∧ Rle (apply_fun r x) 1.
L181352
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r x) Hm1R real_1 HrxI).
L181352
We prove the intermediate claim Hlo: Rle (minus_SNo 1) (apply_fun r x).
L181354
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r x)) (Rle (apply_fun r x) 1) HrxB).
L181354
We prove the intermediate claim Hhi: Rle (apply_fun r x) (minus_SNo one_third).
L181356
An exact proof term for the current goal is (Rlt_implies_Rle (apply_fun r x) (minus_SNo one_third) Hr_lt_left).
L181356
We prove the intermediate claim HrxIleft: apply_fun r x ∈ closed_interval (minus_SNo 1) (minus_SNo one_third).
L181358
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) (minus_SNo one_third) (apply_fun r x) Hm1R Hm13R HrxR Hlo Hhi).
L181359
We prove the intermediate claim HrxV: apply_fun r x ∈ (closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I.
L181361
An exact proof term for the current goal is (binintersectI (closed_interval (minus_SNo 1) (minus_SNo one_third)) I (apply_fun r x) HrxIleft HrxI).
L181361
We prove the intermediate claim HxPre: x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I).
L181363
An exact proof term for the current goal is (SepI A (λz : set ⇒ apply_fun r z ∈ ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I)) x HxA HrxV).
L181364
We prove the intermediate claim HuxEq: apply_fun (u_of r) x = minus_SNo one_third.
L181366
An exact proof term for the current goal is (Hu_left_eq x HxPre).
L181366
We prove the intermediate claim HrNumEqL: apply_fun rNum x = add_SNo (apply_fun r x) one_third.
L181368
rewrite the current goal using HrNumEq (from left to right).
L181368
rewrite the current goal using HuxEq (from left to right).
L181369
rewrite the current goal using (minus_SNo_minus_SNo_R one_third one_third_in_R) (from left to right) at position 1.
Use reflexivity.
L181371
rewrite the current goal using HrNumEqL (from left to right).
L181372
We prove the intermediate claim H13R: one_third ∈ R.
L181374
An exact proof term for the current goal is one_third_in_R.
L181374
We prove the intermediate claim H13S: SNo one_third.
L181376
An exact proof term for the current goal is (real_SNo one_third H13R).
L181376
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
L181378
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L181380
An exact proof term for the current goal is (real_minus_SNo den HdenR).
L181380
We prove the intermediate claim HsumR: add_SNo (apply_fun r x) one_third ∈ R.
L181382
An exact proof term for the current goal is (real_add_SNo (apply_fun r x) HrxR one_third H13R).
L181382
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r x) one_third).
L181384
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r x) one_third Hm1R HrxR H13R Hlo).
L181384
We prove the intermediate claim Hlow: Rle (minus_SNo den) (add_SNo (apply_fun r x) one_third).
L181386
rewrite the current goal using HdenDef (from left to right) at position 1.
L181386
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L181387
An exact proof term for the current goal is Hlow_tmp.
L181388
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r x) one_third) (add_SNo (minus_SNo one_third) one_third).
L181390
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r x) (minus_SNo one_third) one_third HrxR Hm13R H13R Hhi).
L181390
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r x) one_third) 0.
L181392
We prove the intermediate claim HtR: add_SNo (minus_SNo one_third) one_third ∈ R.
L181393
An exact proof term for the current goal is (real_add_SNo (minus_SNo one_third) Hm13R one_third H13R).
L181393
We prove the intermediate claim HtEq0: add_SNo (minus_SNo one_third) one_third = 0.
L181395
An exact proof term for the current goal is (add_SNo_minus_SNo_linv one_third H13S).
L181395
We prove the intermediate claim Ht0: Rle (add_SNo (minus_SNo one_third) one_third) 0.
L181397
rewrite the current goal using HtEq0 (from left to right).
L181397
An exact proof term for the current goal is (Rle_refl 0 real_0).
L181398
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r x) one_third) (add_SNo (minus_SNo one_third) one_third) 0 Hup0_tmp Ht0).
L181401
We prove the intermediate claim H0leDen: Rle 0 den.
L181403
rewrite the current goal using HdenDef (from left to right) at position 1.
L181403
An exact proof term for the current goal is Rle_0_two_thirds.
L181404
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r x) one_third) den.
L181406
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r x) one_third) 0 den Hup0 H0leDen).
L181406
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo den) den (add_SNo (apply_fun r x) one_third) HmdenR HdenR HsumR Hlow Hup).
L181409
Assume Hr_not_lt_left: ¬ (Rlt (apply_fun r x) (minus_SNo one_third)).
L181409
Apply (xm (Rlt one_third (apply_fun r x))) to the current goal.
L181411
Assume Hr_lt_right: Rlt one_third (apply_fun r x).
L181411
We prove the intermediate claim Hu_right_eq: ∀y : set, y ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) y = one_third.
L181415
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 (u_of r) ∧ (∀y0 : set, y0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) y0 = minus_SNo one_third)) (∀y0 : set, y0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) y0 = one_third) Hu_pack).
L181421
We prove the intermediate claim HrxB: Rle (minus_SNo 1) (apply_fun r x) ∧ Rle (apply_fun r x) 1.
L181423
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r x) (real_minus_SNo 1 real_1) real_1 HrxI).
L181423
We prove the intermediate claim Hhi: Rle (apply_fun r x) 1.
L181425
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r x)) (Rle (apply_fun r x) 1) HrxB).
L181425
We prove the intermediate claim Hlo: Rle one_third (apply_fun r x).
L181427
An exact proof term for the current goal is (Rlt_implies_Rle one_third (apply_fun r x) Hr_lt_right).
L181427
We prove the intermediate claim HrxIright: apply_fun r x ∈ closed_interval one_third 1.
L181429
An exact proof term for the current goal is (closed_intervalI_of_Rle one_third 1 (apply_fun r x) one_third_in_R real_1 HrxR Hlo Hhi).
L181430
We prove the intermediate claim HrxV: apply_fun r x ∈ (closed_interval one_third 1) ∩ I.
L181432
An exact proof term for the current goal is (binintersectI (closed_interval one_third 1) I (apply_fun r x) HrxIright HrxI).
L181432
We prove the intermediate claim HxPre: x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I).
L181434
An exact proof term for the current goal is (SepI A (λz : set ⇒ apply_fun r z ∈ ((closed_interval one_third 1) ∩ I)) x HxA HrxV).
L181435
We prove the intermediate claim HuxEq: apply_fun (u_of r) x = one_third.
L181437
An exact proof term for the current goal is (Hu_right_eq x HxPre).
L181437
We prove the intermediate claim HrNumEqR: apply_fun rNum x = add_SNo (apply_fun r x) (minus_SNo one_third).
L181439
rewrite the current goal using HrNumEq (from left to right).
L181439
rewrite the current goal using HuxEq (from left to right).
Use reflexivity.
L181441
rewrite the current goal using HrNumEqR (from left to right).
L181442
We prove the intermediate claim H13R: one_third ∈ R.
L181444
An exact proof term for the current goal is one_third_in_R.
L181444
We prove the intermediate claim H23R: two_thirds ∈ R.
L181446
An exact proof term for the current goal is two_thirds_in_R.
L181446
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L181448
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L181448
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L181450
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L181450
We prove the intermediate claim HrxB: Rle (minus_SNo 1) (apply_fun r x) ∧ Rle (apply_fun r x) 1.
L181452
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r x) (real_minus_SNo 1 real_1) real_1 HrxI).
L181452
We prove the intermediate claim Hrxle1: Rle (apply_fun r x) 1.
L181454
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r x)) (Rle (apply_fun r x) 1) HrxB).
L181454
We prove the intermediate claim HrNumR: add_SNo (apply_fun r x) (minus_SNo one_third) ∈ R.
L181456
An exact proof term for the current goal is (real_add_SNo (apply_fun r x) HrxR (minus_SNo one_third) Hm13R).
L181456
We prove the intermediate claim H0le_rNum_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r x) (minus_SNo one_third)).
L181459
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r x) (minus_SNo one_third) H13R HrxR Hm13R Hlo).
L181459
We prove the intermediate claim H13S: SNo one_third.
L181461
An exact proof term for the current goal is (real_SNo one_third H13R).
L181461
We prove the intermediate claim H0le_rNum: Rle 0 (add_SNo (apply_fun r x) (minus_SNo one_third)).
L181463
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L181463
An exact proof term for the current goal is H0le_rNum_tmp.
L181464
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L181466
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L181466
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
L181468
We prove the intermediate claim Hlow0: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r x) (minus_SNo one_third)).
L181470
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r x) (minus_SNo one_third)) Hm23le0 H0le_rNum).
L181470
We prove the intermediate claim Hlow: Rle (minus_SNo den) (add_SNo (apply_fun r x) (minus_SNo one_third)).
L181472
rewrite the current goal using HdenDef (from left to right) at position 1.
L181472
An exact proof term for the current goal is Hlow0.
L181473
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L181476
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r x) 1 (minus_SNo one_third) HrxR real_1 Hm13R Hrxle1).
L181476
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L181478
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L181478
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L181479
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r x) (minus_SNo one_third)) two_thirds.
L181481
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L181484
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r x) (minus_SNo one_third)) den.
L181486
rewrite the current goal using HdenDef (from left to right) at position 1.
L181486
An exact proof term for the current goal is Hup0.
L181487
We prove the intermediate claim HmdenR: minus_SNo den ∈ R.
L181489
An exact proof term for the current goal is (real_minus_SNo den HdenR).
L181489
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo den) den (add_SNo (apply_fun r x) (minus_SNo one_third)) HmdenR HdenR HrNumR Hlow Hup).
L181493
Assume Hr_not_lt_right: ¬ (Rlt one_third (apply_fun r x)).
L181493
We prove the intermediate claim Hm13R: (minus_SNo one_third) ∈ R.
L181495
An exact proof term for the current goal is (real_minus_SNo one_third one_third_in_R).
L181495
We prove the intermediate claim Hm13_le_rx: Rle (minus_SNo one_third) (apply_fun r x).
L181497
Apply (RleI (minus_SNo one_third) (apply_fun r x) Hm13R HrxR) to the current goal.
L181497
We will prove ¬ (Rlt (apply_fun r x) (minus_SNo one_third)).
L181498
An exact proof term for the current goal is Hr_not_lt_left.
L181499
We prove the intermediate claim Hrx_le_13: Rle (apply_fun r x) one_third.
L181501
Apply (RleI (apply_fun r x) one_third HrxR one_third_in_R) to the current goal.
L181501
We will prove ¬ (Rlt one_third (apply_fun r x)).
L181502
An exact proof term for the current goal is Hr_not_lt_right.
L181503
We prove the intermediate claim HrxI0: apply_fun r x ∈ closed_interval (minus_SNo one_third) one_third.
L181505
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo one_third) one_third (apply_fun r x) Hm13R one_third_in_R HrxR Hm13_le_rx Hrx_le_13).
L181506
rewrite the current goal using HrNumEq (from left to right).
L181507
We prove the intermediate claim H13R: one_third ∈ R.
L181509
An exact proof term for the current goal is one_third_in_R.
L181509
We prove the intermediate claim H23R: two_thirds ∈ R.
L181511
An exact proof term for the current goal is two_thirds_in_R.
L181511
We prove the intermediate claim Hux_bounds: Rle (minus_SNo one_third) (apply_fun (u_of r) x) ∧ Rle (apply_fun (u_of r) x) one_third.
L181513
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun (u_of r) x) Hm13R one_third_in_R HuxI0).
L181514
We prove the intermediate claim Hm13_le_ux: Rle (minus_SNo one_third) (apply_fun (u_of r) x).
L181516
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun (u_of r) x)) (Rle (apply_fun (u_of r) x) one_third) Hux_bounds).
L181517
We prove the intermediate claim Hux_le_13: Rle (apply_fun (u_of r) x) one_third.
L181519
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun (u_of r) x)) (Rle (apply_fun (u_of r) x) one_third) Hux_bounds).
L181520
We prove the intermediate claim Hm13_le_negux: Rle (minus_SNo one_third) (minus_SNo (apply_fun (u_of r) x)).
L181522
An exact proof term for the current goal is (Rle_minus_contra (apply_fun (u_of r) x) one_third Hux_le_13).
L181522
We prove the intermediate claim Hnegux_le_13: Rle (minus_SNo (apply_fun (u_of r) x)) one_third.
L181524
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun (u_of r) x)) (minus_SNo (minus_SNo one_third)).
L181525
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun (u_of r) x) Hm13_le_ux).
L181525
We prove the intermediate claim Hmid: Rle (minus_SNo (minus_SNo one_third)) one_third.
L181527
rewrite the current goal using (minus_SNo_minus_SNo_R one_third one_third_in_R) (from left to right) at position 1.
L181527
An exact proof term for the current goal is (Rle_refl one_third one_third_in_R).
L181528
An exact proof term for the current goal is (Rle_tra (minus_SNo (apply_fun (u_of r) x)) (minus_SNo (minus_SNo one_third)) one_third Htmp Hmid).
L181532
We prove the intermediate claim Hm13R2: minus_SNo one_third ∈ R.
L181534
An exact proof term for the current goal is Hm13R.
L181534
We prove the intermediate claim HneguxR: minus_SNo (apply_fun (u_of r) x) ∈ R.
L181536
An exact proof term for the current goal is (real_minus_SNo (apply_fun (u_of r) x) HuxR).
L181536
We prove the intermediate claim HrNumR: add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x)) ∈ R.
L181538
An exact proof term for the current goal is (real_add_SNo (apply_fun r x) HrxR (minus_SNo (apply_fun (u_of r) x)) HneguxR).
L181538
We prove the intermediate claim Hlow0a: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun (u_of r) x))).
L181541
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun (u_of r) x)) Hm13R2 Hm13R2 HneguxR Hm13_le_negux).
L181542
We prove the intermediate claim Hlow0b: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun (u_of r) x))) (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))).
L181545
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r x) (minus_SNo (apply_fun (u_of r) x)) Hm13R2 HrxR HneguxR Hm13_le_rx).
L181546
We prove the intermediate claim Hlow0: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))).
L181549
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun (u_of r) x))) (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))) Hlow0a Hlow0b).
L181552
We prove the intermediate claim Hlow2: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))).
L181555
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L181555
An exact proof term for the current goal is Hlow0.
L181556
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
L181558
We prove the intermediate claim Hlow: Rle (minus_SNo den) (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))).
L181561
rewrite the current goal using HdenDef (from left to right) at position 1.
L181561
An exact proof term for the current goal is Hlow2.
L181562
We prove the intermediate claim Hup0a: Rle (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))) (add_SNo (apply_fun r x) one_third).
L181565
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r x) (minus_SNo (apply_fun (u_of r) x)) one_third HrxR HneguxR H13R Hnegux_le_13).
L181566
We prove the intermediate claim Hup0b: Rle (add_SNo (apply_fun r x) one_third) (add_SNo one_third one_third).
L181568
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r x) one_third one_third HrxR H13R H13R Hrx_le_13).
L181568
We prove the intermediate claim Hup0c: Rle (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))) (add_SNo one_third one_third).
L181570
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))) (add_SNo (apply_fun r x) one_third) (add_SNo one_third one_third) Hup0a Hup0b).
L181573
We prove the intermediate claim Htwo_def: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L181575
We prove the intermediate claim Hup1: Rle (add_SNo one_third one_third) two_thirds.
L181577
rewrite the current goal using Htwo_def (from right to left) at position 1.
L181577
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L181578
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))) two_thirds.
L181580
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))) (add_SNo one_third one_third) two_thirds Hup0c Hup1).
L181583
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))) den.
L181585
rewrite the current goal using HdenDef (from left to right) at position 1.
L181585
An exact proof term for the current goal is Hup2.
L181586
We prove the intermediate claim HmdenR: minus_SNo den ∈ R.
L181588
An exact proof term for the current goal is (real_minus_SNo den HdenR).
L181588
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo den) den (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))) HmdenR HdenR HrNumR Hlow Hup).
L181591
We prove the intermediate claim HrNumR: apply_fun rNum x ∈ R.
L181593
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun rNum x) HrNumI2).
L181593
We prove the intermediate claim HrNewEq: apply_fun rNew x = div_SNo (apply_fun rNum x) den.
L181595
An exact proof term for the current goal is (compose_div_const_fun_apply A rNum den x HxA HdenR HrNumR).
L181595
rewrite the current goal using HrNewEq (from left to right).
L181596
An exact proof term for the current goal is (div_SNo_closed_interval_scale den (apply_fun rNum x) HdenR HdenPos HrNumI2).
L181597
We prove the intermediate claim HrNew_contI': continuous_map A Ta I (subspace_topology R R_standard_topology I) rNew.
L181599
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology rNew I HrNew_contR HIcR HrNew_img).
L181599
rewrite the current goal using HTiEq (from left to right).
L181600
An exact proof term for the current goal is HrNew_contI'.
L181601
Let n be given.
L181602
Assume HnO: n ∈ ω.
L181602
An exact proof term for the current goal is (HInv_nat n (omega_nat_p n HnO)).
L181603
We prove the intermediate claim HInv_g_R_A: ∀n : set, n ∈ ω → ∀x : set, x ∈ A → apply_fun ((State n) 0) x ∈ R.
(*** Step II invariant on A: each g_n takes values in R ***)
L181608
We prove the intermediate claim HInv_nat: ∀n : set, nat_p n → ∀x : set, x ∈ A → apply_fun ((State n) 0) x ∈ R.
L181610
Apply nat_ind to the current goal.
L181611
Let x be given.
L181611
Assume HxA: x ∈ A.
L181611
We will prove apply_fun ((State 0) 0) x ∈ R.
L181612
We prove the intermediate claim HxX: x ∈ X.
L181614
An exact proof term for the current goal is (HAsubX x HxA).
L181614
We prove the intermediate claim H0: State 0 = BaseState.
L181616
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L181616
rewrite the current goal using H0 (from left to right).
L181617
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L181619
rewrite the current goal using HBase (from left to right).
L181620
rewrite the current goal using (tuple_2_0_eq g0g (f1s,den)) (from left to right).
L181621
rewrite the current goal using (apply_fun_graph X (λx0 : set ⇒ apply_fun g0 x0) x HxX) (from left to right).
L181622
We prove the intermediate claim Hg0I0: apply_fun g0 x ∈ I0.
L181624
An exact proof term for the current goal is (Hfung0 x HxX).
L181624
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun g0 x) Hg0I0).
L181626
Let k be given.
L181626
Assume HkNat: nat_p k.
L181626
Assume IH: ∀x : set, x ∈ A → apply_fun ((State k) 0) x ∈ R.
L181627
Let x be given.
L181628
Assume HxA: x ∈ A.
L181628
We will prove apply_fun ((State (ordsucc k)) 0) x ∈ R.
L181629
We prove the intermediate claim HS: State (ordsucc k) = StepState k (State k).
L181631
An exact proof term for the current goal is (nat_primrec_S BaseState StepState k HkNat).
L181631
rewrite the current goal using HS (from left to right).
L181632
We prove the intermediate claim HkO: k ∈ ω.
L181634
An exact proof term for the current goal is (nat_p_omega k HkNat).
L181634
We prove the intermediate claim HxX: x ∈ X.
L181636
An exact proof term for the current goal is (HAsubX x HxA).
L181636
Set st to be the term State k.
L181637
Set g to be the term st 0.
L181638
Set rpack to be the term st 1.
L181639
Set r to be the term rpack 0.
L181640
Set c to be the term rpack 1.
L181641
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L181644
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L181649
Set cNew to be the term mul_SNo c den.
L181650
We prove the intermediate claim Hdef: StepState k st = (gNew,(rNew,cNew)).
Use reflexivity.
L181652
We prove the intermediate claim HgStep: (StepState k st) 0 = gNew.
L181654
rewrite the current goal using Hdef (from left to right).
L181654
An exact proof term for the current goal is (tuple_2_0_eq gNew (rNew,cNew)).
L181655
rewrite the current goal using HgStep (from left to right).
L181656
We prove the intermediate claim HgEq: g = (State k) 0.
Use reflexivity.
L181658
We prove the intermediate claim HgxR: apply_fun g x ∈ R.
L181660
rewrite the current goal using HgEq (from left to right).
L181660
An exact proof term for the current goal is (IH x HxA).
L181661
We prove the intermediate claim HrEq: r = ((State k) 1) 0.
Use reflexivity.
L181663
We prove the intermediate claim Hr_contI: continuous_map A Ta I Ti r.
L181665
rewrite the current goal using HrEq (from left to right).
L181665
An exact proof term for the current goal is (HInv_r_contI k HkO).
L181666
We prove the intermediate claim Hu_pack: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x0 = one_third).
L181673
An exact proof term for the current goal is (Hu_of_prop r Hr_contI).
L181673
We prove the intermediate claim Hu_contI0: continuous_map X Tx I0 T0 (u_of r).
L181675
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r)) (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third) (andEL ((continuous_map X Tx I0 T0 (u_of r)) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x0 = one_third) Hu_pack)).
L181685
We prove the intermediate claim Hu_fun: function_on (u_of r) X I0.
L181687
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 (u_of r) Hu_contI0).
L181687
We prove the intermediate claim HuxI0: apply_fun (u_of r) x ∈ I0.
L181689
An exact proof term for the current goal is (Hu_fun x HxX).
L181689
We prove the intermediate claim HuxR: apply_fun (u_of r) x ∈ R.
L181691
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun (u_of r) x) HuxI0).
L181691
We prove the intermediate claim HcEq: c = ((State k) 1) 1.
Use reflexivity.
L181693
We prove the intermediate claim HcR: c ∈ R.
L181695
rewrite the current goal using HcEq (from left to right).
L181695
An exact proof term for the current goal is (andEL (((State k) 1) 1 ∈ R) (0 < ((State k) 1) 1) (HInv_cpos k HkO)).
L181699
We prove the intermediate claim HcorrEq: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x = mul_SNo (apply_fun (u_of r) x) c.
L181703
We prove the intermediate claim Hcomp: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x = apply_fun (mul_const_fun c) (apply_fun (u_of r) x).
L181706
An exact proof term for the current goal is (compose_fun_apply X (u_of r) (mul_const_fun c) x HxX).
L181706
rewrite the current goal using Hcomp (from left to right).
L181707
An exact proof term for the current goal is (mul_const_fun_apply c (apply_fun (u_of r) x) HcR HuxR).
L181708
We prove the intermediate claim HcorrR: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x ∈ R.
L181710
rewrite the current goal using HcorrEq (from left to right).
L181710
An exact proof term for the current goal is (real_mul_SNo (apply_fun (u_of r) x) HuxR c HcR).
L181711
We prove the intermediate claim HgNewEval: apply_fun gNew x = add_SNo (apply_fun g x) (apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x).
L181716
An exact proof term for the current goal is (add_of_pair_map_apply X g (compose_fun X (u_of r) (mul_const_fun c)) x HxX HgxR HcorrR).
L181716
rewrite the current goal using HgNewEval (from left to right).
L181717
An exact proof term for the current goal is (real_add_SNo (apply_fun g x) HgxR (apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x) HcorrR).
L181719
Let n be given.
L181720
Assume HnO: n ∈ ω.
L181720
An exact proof term for the current goal is (HInv_nat n (omega_nat_p n HnO)).
L181721
We prove the intermediate claim HInv_residual_identity_A: ∀n : set, n ∈ ω → ∀x : set, x ∈ A → add_SNo (apply_fun ((State n) 0) x) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) = apply_fun f x.
(*** Step II invariant on A: f(x) = g_n(x) + c_n times r_n(x) ***)
L181729
We prove the intermediate claim HInv_nat: ∀n : set, nat_p n → ∀x : set, x ∈ A → add_SNo (apply_fun ((State n) 0) x) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) = apply_fun f x.
L181733
Apply nat_ind to the current goal.
L181734
Let x be given.
L181734
Assume HxA: x ∈ A.
L181734
We will prove add_SNo (apply_fun ((State 0) 0) x) (mul_SNo (apply_fun (((State 0) 1) 0) x) (((State 0) 1) 1)) = apply_fun f x.
L181737
We prove the intermediate claim HxX: x ∈ X.
L181739
An exact proof term for the current goal is (HAsubX x HxA).
L181739
We prove the intermediate claim H0: State 0 = BaseState.
L181741
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L181741
rewrite the current goal using H0 (from left to right).
L181742
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L181744
rewrite the current goal using HBase (from left to right).
L181745
rewrite the current goal using (tuple_2_0_eq g0g (f1s,den)) (from left to right) at position 1.
L181746
rewrite the current goal using (tuple_2_1_eq g0g (f1s,den)) (from left to right) at position 1.
L181747
rewrite the current goal using (tuple_2_0_eq f1s den) (from left to right) at position 1.
L181748
rewrite the current goal using (tuple_2_1_eq g0g (f1s,den)) (from left to right) at position 1.
L181749
rewrite the current goal using (tuple_2_1_eq f1s den) (from left to right) at position 1.
L181750
We prove the intermediate claim Hg0g_app: apply_fun g0g x = apply_fun g0 x.
L181752
rewrite the current goal using (apply_fun_graph X (λx0 : set ⇒ apply_fun g0 x0) x HxX) (from left to right).
Use reflexivity.
L181753
rewrite the current goal using Hg0g_app (from left to right) at position 1.
L181754
We prove the intermediate claim Hf1s_app: apply_fun f1s x = div_SNo (apply_fun f1 x) den.
L181756
An exact proof term for the current goal is (Hf1s_apply x HxA).
L181756
rewrite the current goal using Hf1s_app (from left to right) at position 1.
L181757
We prove the intermediate claim Hf1xI2: apply_fun f1 x ∈ I2.
L181759
An exact proof term for the current goal is (Hf1_range x HxA).
L181759
We prove the intermediate claim Hf1xI2': apply_fun f1 x ∈ closed_interval (minus_SNo den) den.
L181761
We prove the intermediate claim HI2def: I2 = closed_interval (minus_SNo den) den.
Use reflexivity.
L181762
rewrite the current goal using HI2def (from left to right).
L181763
An exact proof term for the current goal is Hf1xI2.
L181764
We prove the intermediate claim Hf1xR: apply_fun f1 x ∈ R.
L181766
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun f1 x) Hf1xI2').
L181766
We prove the intermediate claim Hf1xS: SNo (apply_fun f1 x).
L181768
An exact proof term for the current goal is (real_SNo (apply_fun f1 x) Hf1xR).
L181768
We prove the intermediate claim HmulDiv: mul_SNo (div_SNo (apply_fun f1 x) den) den = apply_fun f1 x.
L181770
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun f1 x) den Hf1xS H23S H23ne0).
L181770
rewrite the current goal using HmulDiv (from left to right) at position 1.
L181771
rewrite the current goal using (Hf1_apply x HxA) (from left to right) at position 1.
L181772
We prove the intermediate claim HfxR: apply_fun f x ∈ R.
L181774
An exact proof term for the current goal is (Hf_R x HxA).
L181774
We prove the intermediate claim HfxS: SNo (apply_fun f x).
L181776
An exact proof term for the current goal is (real_SNo (apply_fun f x) HfxR).
L181776
We prove the intermediate claim Hg0I0: apply_fun g0 x ∈ I0.
L181778
An exact proof term for the current goal is (Hfung0 x HxX).
L181778
We prove the intermediate claim Hg0R: apply_fun g0 x ∈ R.
L181780
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun g0 x) Hg0I0).
L181780
We prove the intermediate claim Hg0S: SNo (apply_fun g0 x).
L181782
An exact proof term for the current goal is (real_SNo (apply_fun g0 x) Hg0R).
L181782
rewrite the current goal using (add_SNo_assoc (apply_fun g0 x) (apply_fun f x) (minus_SNo (apply_fun g0 x)) Hg0S HfxS (SNo_minus_SNo (apply_fun g0 x) Hg0S)) (from left to right) at position 1.
L181784
rewrite the current goal using (add_SNo_com (apply_fun g0 x) (apply_fun f x) Hg0S HfxS) (from left to right) at position 1.
L181785
rewrite the current goal using (add_SNo_assoc (apply_fun f x) (apply_fun g0 x) (minus_SNo (apply_fun g0 x)) HfxS Hg0S (SNo_minus_SNo (apply_fun g0 x) Hg0S)) (from right to left) at position 1.
L181787
rewrite the current goal using (add_SNo_minus_SNo_rinv (apply_fun g0 x) Hg0S) (from left to right) at position 1.
L181788
rewrite the current goal using (add_SNo_0R (apply_fun f x) HfxS) (from left to right) at position 1.
Use reflexivity.
L181791
Let k be given.
L181791
Assume HkNat: nat_p k.
L181791
Assume IH: ∀x : set, x ∈ A → add_SNo (apply_fun ((State k) 0) x) (mul_SNo (apply_fun (((State k) 1) 0) x) (((State k) 1) 1)) = apply_fun f x.
L181795
Let x be given.
L181796
Assume HxA: x ∈ A.
L181796
We will prove add_SNo (apply_fun ((State (ordsucc k)) 0) x) (mul_SNo (apply_fun (((State (ordsucc k)) 1) 0) x) (((State (ordsucc k)) 1) 1)) = apply_fun f x.
L181799
We prove the intermediate claim HS: State (ordsucc k) = StepState k (State k).
L181801
An exact proof term for the current goal is (nat_primrec_S BaseState StepState k HkNat).
L181801
rewrite the current goal using HS (from left to right).
L181802
rewrite the current goal using (IH x HxA) (from right to left).
L181803
We prove the intermediate claim HkO: k ∈ ω.
L181805
An exact proof term for the current goal is (nat_p_omega k HkNat).
L181805
We prove the intermediate claim HxX: x ∈ X.
L181807
An exact proof term for the current goal is (HAsubX x HxA).
L181807
Set st to be the term State k.
L181808
Set g to be the term st 0.
L181809
Set rpack to be the term st 1.
L181810
Set r to be the term rpack 0.
L181811
Set c to be the term rpack 1.
L181812
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L181815
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L181820
Set cNew to be the term mul_SNo c den.
L181821
We prove the intermediate claim Hdef: StepState k st = (gNew,(rNew,cNew)).
Use reflexivity.
L181823
We prove the intermediate claim HgStep: (StepState k st) 0 = gNew.
L181825
rewrite the current goal using Hdef (from left to right).
L181825
An exact proof term for the current goal is (tuple_2_0_eq gNew (rNew,cNew)).
L181826
We prove the intermediate claim HrpackStep: (StepState k st) 1 = (rNew,cNew).
L181828
rewrite the current goal using Hdef (from left to right).
L181828
An exact proof term for the current goal is (tuple_2_1_eq gNew (rNew,cNew)).
L181829
We prove the intermediate claim HrStep: ((StepState k st) 1) 0 = rNew.
L181831
rewrite the current goal using HrpackStep (from left to right).
L181831
An exact proof term for the current goal is (tuple_2_0_eq rNew cNew).
L181832
We prove the intermediate claim HcStep: ((StepState k st) 1) 1 = cNew.
L181834
rewrite the current goal using HrpackStep (from left to right).
L181834
An exact proof term for the current goal is (tuple_2_1_eq rNew cNew).
L181835
rewrite the current goal using HgStep (from left to right) at position 1.
L181836
rewrite the current goal using HrStep (from left to right) at position 1.
L181837
rewrite the current goal using HcStep (from left to right) at position 1.
L181838
Set gx to be the term apply_fun g x.
L181839
Set rx to be the term apply_fun r x.
L181840
Set ux to be the term apply_fun (u_of r) x.
L181841
We prove the intermediate claim HgxR: gx ∈ R.
L181843
We prove the intermediate claim HgEq: g = (State k) 0.
Use reflexivity.
L181844
rewrite the current goal using HgEq (from left to right).
L181845
An exact proof term for the current goal is (HInv_g_R_A k HkO x HxA).
L181846
We prove the intermediate claim HgxS: SNo gx.
L181848
An exact proof term for the current goal is (real_SNo gx HgxR).
L181848
We prove the intermediate claim HrEq: r = ((State k) 1) 0.
Use reflexivity.
L181850
We prove the intermediate claim Hr_contI: continuous_map A Ta I Ti r.
L181852
rewrite the current goal using HrEq (from left to right).
L181852
An exact proof term for the current goal is (HInv_r_contI k HkO).
L181853
We prove the intermediate claim Hr_fun: function_on r A I.
L181855
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r Hr_contI).
L181855
We prove the intermediate claim HrxI: rx ∈ I.
L181857
An exact proof term for the current goal is (Hr_fun x HxA).
L181857
We prove the intermediate claim HrxR: rx ∈ R.
L181859
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 rx HrxI).
L181859
We prove the intermediate claim HrxS: SNo rx.
L181861
An exact proof term for the current goal is (real_SNo rx HrxR).
L181861
We prove the intermediate claim Hu_pack: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x0 = one_third).
L181868
An exact proof term for the current goal is (Hu_of_prop r Hr_contI).
L181868
We prove the intermediate claim Hu_contI0: continuous_map X Tx I0 T0 (u_of r).
L181870
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r)) (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third) (andEL ((continuous_map X Tx I0 T0 (u_of r)) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x0 = one_third) Hu_pack)).
L181880
We prove the intermediate claim Hu_fun: function_on (u_of r) X I0.
L181882
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 (u_of r) Hu_contI0).
L181882
We prove the intermediate claim HuxI0: ux ∈ I0.
L181884
An exact proof term for the current goal is (Hu_fun x HxX).
L181884
We prove the intermediate claim HuxR: ux ∈ R.
L181886
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third ux HuxI0).
L181886
We prove the intermediate claim HuxS: SNo ux.
L181888
An exact proof term for the current goal is (real_SNo ux HuxR).
L181888
We prove the intermediate claim HcEq: c = ((State k) 1) 1.
Use reflexivity.
L181890
We prove the intermediate claim HcR: c ∈ R.
L181892
rewrite the current goal using HcEq (from left to right).
L181892
An exact proof term for the current goal is (andEL (((State k) 1) 1 ∈ R) (0 < ((State k) 1) 1) (HInv_cpos k HkO)).
L181896
We prove the intermediate claim HcS: SNo c.
L181898
An exact proof term for the current goal is (real_SNo c HcR).
L181898
We prove the intermediate claim HcorrEq: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x = mul_SNo ux c.
L181901
We prove the intermediate claim Hcomp: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x = apply_fun (mul_const_fun c) ux.
L181904
An exact proof term for the current goal is (compose_fun_apply X (u_of r) (mul_const_fun c) x HxX).
L181904
rewrite the current goal using Hcomp (from left to right).
L181905
An exact proof term for the current goal is (mul_const_fun_apply c ux HcR HuxR).
L181906
We prove the intermediate claim HcorrR: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x ∈ R.
L181908
rewrite the current goal using HcorrEq (from left to right).
L181908
An exact proof term for the current goal is (real_mul_SNo ux HuxR c HcR).
L181909
We prove the intermediate claim HgNewEval: apply_fun gNew x = add_SNo gx (apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x).
L181913
An exact proof term for the current goal is (add_of_pair_map_apply X g (compose_fun X (u_of r) (mul_const_fun c)) x HxX HgxR HcorrR).
L181913
rewrite the current goal using HgNewEval (from left to right).
L181914
rewrite the current goal using HcorrEq (from left to right) at position 1.
L181915
Set uxc to be the term mul_SNo ux c.
L181916
We prove the intermediate claim HuxcR: uxc ∈ R.
L181918
An exact proof term for the current goal is (real_mul_SNo ux HuxR c HcR).
L181918
We prove the intermediate claim HuxcS: SNo uxc.
L181920
An exact proof term for the current goal is (real_SNo uxc HuxcR).
L181920
Set rNum to be the term compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R.
L181921
We prove the intermediate claim HrNumEval: apply_fun rNum x = add_SNo rx (minus_SNo ux).
L181923
An exact proof term for the current goal is (add_of_pair_map_neg_apply A r (u_of r) x HxA HrxR HuxR).
L181923
We prove the intermediate claim HrNumR: apply_fun rNum x ∈ R.
L181925
rewrite the current goal using HrNumEval (from left to right).
L181925
An exact proof term for the current goal is (real_add_SNo rx HrxR (minus_SNo ux) (real_minus_SNo ux HuxR)).
L181926
We prove the intermediate claim HrNewEq: apply_fun rNew x = div_SNo (apply_fun rNum x) den.
L181928
An exact proof term for the current goal is (compose_div_const_fun_apply A rNum den x HxA HdenR HrNumR).
L181928
rewrite the current goal using HrNewEq (from left to right).
L181929
rewrite the current goal using HrNumEval (from left to right) at position 1.
L181930
Set num to be the term add_SNo rx (minus_SNo ux).
L181931
We prove the intermediate claim HnumDef: num = add_SNo rx (minus_SNo ux).
Use reflexivity.
L181933
rewrite the current goal using HnumDef (from right to left) at position 1.
L181934
We prove the intermediate claim HnumR: num ∈ R.
L181936
An exact proof term for the current goal is (real_add_SNo rx HrxR (minus_SNo ux) (real_minus_SNo ux HuxR)).
L181936
We prove the intermediate claim HnumS: SNo num.
L181938
An exact proof term for the current goal is (real_SNo num HnumR).
L181938
We prove the intermediate claim HdenS: SNo den.
L181940
An exact proof term for the current goal is (real_SNo den HdenR).
L181940
We prove the intermediate claim HdivR: div_SNo num den ∈ R.
L181942
An exact proof term for the current goal is (real_div_SNo num HnumR den HdenR).
L181942
We prove the intermediate claim HdivS: SNo (div_SNo num den).
L181944
An exact proof term for the current goal is (real_SNo (div_SNo num den) HdivR).
L181944
We prove the intermediate claim HcNewDef: cNew = mul_SNo c den.
Use reflexivity.
(*** reduce the correction term algebraically ***)
L181947
rewrite the current goal using HcNewDef (from left to right).
L181948
We prove the intermediate claim HmulAssoc: mul_SNo (div_SNo num den) (mul_SNo c den) = mul_SNo (mul_SNo (div_SNo num den) c) den.
L181951
An exact proof term for the current goal is (mul_SNo_assoc (div_SNo num den) c den HdivS HcS HdenS).
L181951
rewrite the current goal using HmulAssoc (from left to right).
L181952
We prove the intermediate claim HmulSwap: mul_SNo (mul_SNo (div_SNo num den) c) den = mul_SNo (mul_SNo (div_SNo num den) den) c.
L181955
An exact proof term for the current goal is (mul_SNo_com_3b_1_2 (div_SNo num den) c den HdivS HcS HdenS).
L181955
rewrite the current goal using HmulSwap (from left to right).
L181956
We prove the intermediate claim Hdenne0: den ≠ 0.
L181958
An exact proof term for the current goal is H23ne0.
L181958
We prove the intermediate claim Hcancel: mul_SNo (div_SNo num den) den = num.
L181960
An exact proof term for the current goal is (mul_div_SNo_invL num den HnumS HdenS Hdenne0).
L181960
rewrite the current goal using Hcancel (from left to right).
L181961
We prove the intermediate claim HnumEq: num = add_SNo rx (minus_SNo ux).
Use reflexivity.
L181963
rewrite the current goal using HnumEq (from left to right).
L181964
We prove the intermediate claim HuxcDef: uxc = mul_SNo ux c.
Use reflexivity.
L181966
rewrite the current goal using HuxcDef (from left to right) at position 1.
L181967
An exact proof term for the current goal is (Tietze_stepII_algebra_tail gx rx ux c HgxS HrxS HuxS HcS).
L181968
Let n be given.
L181969
Assume HnO: n ∈ ω.
L181969
Let x be given.
L181970
Assume HxA: x ∈ A.
L181970
An exact proof term for the current goal is (HInv_nat n (omega_nat_p n HnO) x HxA).
L181971
We prove the intermediate claim HInv_g_FS: ∀n : set, n ∈ ω → (State n) 0 ∈ function_space X R.
(*** The remaining proof obligations are split so they can be refined independently later. ***)
(*** helper: State n yields a real-valued map on X (in function_space X R) ***)
L181975
We prove the intermediate claim Hfun_g0: function_on g0 X R.
L181976
An exact proof term for the current goal is (continuous_map_function_on X Tx R R_standard_topology g0 Hg0contR).
L181976
We prove the intermediate claim Hg0g_FS: g0g ∈ function_space X R.
L181978
Set gg to be the term (λx : set ⇒ apply_fun g0 x).
L181978
We prove the intermediate claim Hgg: ∀x : set, x ∈ X → gg x ∈ R.
L181980
Let x be given.
L181980
Assume HxX: x ∈ X.
L181980
An exact proof term for the current goal is (Hfun_g0 x HxX).
L181981
We prove the intermediate claim Hg0g_def: g0g = graph X gg.
Use reflexivity.
L181983
rewrite the current goal using Hg0g_def (from left to right).
L181984
An exact proof term for the current goal is (graph_in_function_space X R gg Hgg).
L181985
We prove the intermediate claim HInv_nat: ∀n : set, nat_p n → (State n) 0 ∈ function_space X R.
L181987
Apply nat_ind to the current goal.
L181988
We will prove (State 0) 0 ∈ function_space X R.
L181989
We prove the intermediate claim H0: State 0 = BaseState.
(*** base case ***)
L181991
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L181991
rewrite the current goal using H0 (from left to right).
L181992
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L181994
rewrite the current goal using HBase (from left to right).
L181995
rewrite the current goal using (tuple_2_0_eq g0g (f1s,den)) (from left to right).
L181996
An exact proof term for the current goal is Hg0g_FS.
L181998
Let n be given.
L181998
Assume HnNat: nat_p n.
L181998
Assume IH: (State n) 0 ∈ function_space X R.
L181999
We will prove (State (ordsucc n)) 0 ∈ function_space X R.
L182000
We prove the intermediate claim HS: State (ordsucc n) = StepState n (State n).
L182002
An exact proof term for the current goal is (nat_primrec_S BaseState StepState n HnNat).
L182002
rewrite the current goal using HS (from left to right).
L182003
Set st to be the term State n.
L182004
Set g to be the term st 0.
L182005
Set rpack to be the term st 1.
L182006
Set r to be the term rpack 0.
L182007
Set c to be the term rpack 1.
L182008
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L182011
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L182016
Set cNew to be the term mul_SNo c den.
L182017
We prove the intermediate claim HStep0: (StepState n st) 0 = gNew.
L182019
We prove the intermediate claim Hdef: StepState n st = (gNew,(rNew,cNew)).
Use reflexivity.
L182020
rewrite the current goal using Hdef (from left to right).
L182021
An exact proof term for the current goal is (tuple_2_0_eq gNew (rNew,cNew)).
L182022
rewrite the current goal using HStep0 (from left to right).
L182023
We prove the intermediate claim Hg_on: function_on g X R.
L182025
An exact proof term for the current goal is (function_on_of_function_space g X R IH).
L182025
We prove the intermediate claim HnO: n ∈ ω.
L182027
An exact proof term for the current goal is (nat_p_omega n HnNat).
L182027
We prove the intermediate claim Hcpack: ((State n) 1) 1 ∈ R ∧ 0 < ((State n) 1) 1.
L182029
An exact proof term for the current goal is (HInv_cpos n HnO).
L182029
We prove the intermediate claim Hc_eq: c = ((State n) 1) 1.
Use reflexivity.
L182031
We prove the intermediate claim HcR0: ((State n) 1) 1 ∈ R.
L182033
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) Hcpack).
L182033
We prove the intermediate claim HcR: c ∈ R.
L182035
rewrite the current goal using Hc_eq (from left to right).
L182035
An exact proof term for the current goal is HcR0.
L182035
We prove the intermediate claim Hr_contI: continuous_map A Ta I Ti r.
L182037
An exact proof term for the current goal is (HInv_r_contI n HnO).
L182037
We prove the intermediate claim Hu_pack: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x = one_third).
L182044
An exact proof term for the current goal is (Hu_of_prop r Hr_contI).
L182044
We prove the intermediate claim Hu_left: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third).
L182049
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x = one_third) Hu_pack).
L182055
We prove the intermediate claim Hu_cont: continuous_map X Tx I0 T0 (u_of r).
L182057
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r)) (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third) Hu_left).
L182061
We prove the intermediate claim Hu_fun_I0: function_on (u_of r) X I0.
L182063
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 (u_of r) Hu_cont).
L182063
We prove the intermediate claim HI0SubR: I0 ⊆ R.
L182065
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L182065
We prove the intermediate claim Hu_fun_R: function_on (u_of r) X R.
L182067
Let x be given.
L182067
Assume HxX: x ∈ X.
L182067
An exact proof term for the current goal is (HI0SubR (apply_fun (u_of r) x) (Hu_fun_I0 x HxX)).
L182068
We prove the intermediate claim HmulFS: mul_const_fun c ∈ function_space R R.
L182070
An exact proof term for the current goal is (mul_const_fun_in_function_space c HcR).
L182070
We prove the intermediate claim Hmul_on_R: function_on (mul_const_fun c) R R.
L182072
An exact proof term for the current goal is (function_on_of_function_space (mul_const_fun c) R R HmulFS).
L182072
We prove the intermediate claim Hmul_on_I0: function_on (mul_const_fun c) I0 R.
L182074
An exact proof term for the current goal is (function_on_restrict_dom (mul_const_fun c) R I0 R Hmul_on_R HI0SubR).
L182074
We prove the intermediate claim Hh_on: function_on (compose_fun X (u_of r) (mul_const_fun c)) X R.
L182076
An exact proof term for the current goal is (function_on_compose_fun X I0 R (u_of r) (mul_const_fun c) Hu_fun_I0 Hmul_on_I0).
L182076
We prove the intermediate claim Hpair_FS: pair_map X g (compose_fun X (u_of r) (mul_const_fun c)) ∈ function_space X (setprod R R).
L182078
An exact proof term for the current goal is (pair_map_in_function_space X R R g (compose_fun X (u_of r) (mul_const_fun c)) Hg_on Hh_on).
L182078
We prove the intermediate claim Hpair_on: function_on (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) X (setprod R R).
L182080
An exact proof term for the current goal is (function_on_of_function_space (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) X (setprod R R) Hpair_FS).
L182080
We prove the intermediate claim HaddFS: add_fun_R ∈ function_space (setprod R R) R.
L182082
An exact proof term for the current goal is add_fun_R_in_function_space.
L182082
We prove the intermediate claim Hadd_on: function_on add_fun_R (setprod R R) R.
L182084
An exact proof term for the current goal is (function_on_of_function_space add_fun_R (setprod R R) R HaddFS).
L182084
An exact proof term for the current goal is (compose_fun_in_function_space X (setprod R R) R (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R Hpair_on Hadd_on).
L182087
Let n be given.
L182088
Assume HnO: n ∈ ω.
L182088
An exact proof term for the current goal is (HInv_nat n (omega_nat_p n HnO)).
L182089
Apply andI to the current goal.
L182091
We will prove (((function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I)) ∧ (∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x))).
L182099
Apply andI to the current goal.
L182101
We will prove ((function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I)).
L182104
Apply andI to the current goal.
L182106
We will prove (function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))).
L182107
Apply andI to the current goal.
L182109
Let n be given.
L182110
Assume HnO: n ∈ ω.
L182110
We will prove apply_fun fn n ∈ function_space X R.
(*** fn n lies in function_space X R (invariant on State n) ***)
L182111
We prove the intermediate claim Hdef: fn = graph ω (λn0 : set ⇒ (State n0) 0).
Use reflexivity.
L182113
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λn0 : set ⇒ (State n0) 0) n Hdef HnO) (from left to right).
L182114
An exact proof term for the current goal is (HInv_g_FS n HnO).
L182116
We prove the intermediate claim HInv_g_cont: ∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology ((State n) 0).
(*** continuity of each fn n (invariant on State n) ***)
L182118
We prove the intermediate claim Hg0g_cont: continuous_map X Tx R R_standard_topology g0g.
L182119
We prove the intermediate claim HTx': topology_on X Tx.
L182120
An exact proof term for the current goal is (continuous_map_topology_dom X Tx R R_standard_topology g0 Hg0contR).
L182120
We prove the intermediate claim HTR: topology_on R R_standard_topology.
L182122
An exact proof term for the current goal is (continuous_map_topology_cod X Tx R R_standard_topology g0 Hg0contR).
L182122
We prove the intermediate claim Hfun_g0: function_on g0 X R.
L182124
An exact proof term for the current goal is (continuous_map_function_on X Tx R R_standard_topology g0 Hg0contR).
L182124
We prove the intermediate claim Hfun_g0g: function_on g0g X R.
L182126
Let x be given.
L182126
Assume HxX: x ∈ X.
L182126
We will prove apply_fun g0g x ∈ R.
L182127
rewrite the current goal using (apply_fun_graph X (λx0 : set ⇒ apply_fun g0 x0) x HxX) (from left to right) at position 1.
L182128
An exact proof term for the current goal is (Hfun_g0 x HxX).
L182129
We prove the intermediate claim Hpre_g0g: ∀V : set, V ∈ R_standard_topology → preimage_of X g0g V ∈ Tx.
L182131
Let V be given.
L182131
Assume HV: V ∈ R_standard_topology.
L182131
We prove the intermediate claim Hpre_g0: preimage_of X g0 V ∈ Tx.
L182133
An exact proof term for the current goal is (continuous_map_preimage X Tx R R_standard_topology g0 Hg0contR V HV).
L182133
We prove the intermediate claim Heq: preimage_of X g0g V = preimage_of X g0 V.
L182135
Apply set_ext to the current goal.
L182136
Let x be given.
L182136
Assume Hx: x ∈ preimage_of X g0g V.
L182136
We will prove x ∈ preimage_of X g0 V.
L182137
We prove the intermediate claim HxX: x ∈ X.
L182139
An exact proof term for the current goal is (SepE1 X (λy : set ⇒ apply_fun g0g y ∈ V) x Hx).
L182139
We prove the intermediate claim Himg: apply_fun g0g x ∈ V.
L182141
An exact proof term for the current goal is (SepE2 X (λy : set ⇒ apply_fun g0g y ∈ V) x Hx).
L182141
We prove the intermediate claim Himg': apply_fun g0 x ∈ V.
L182143
rewrite the current goal using (apply_fun_graph X (λx0 : set ⇒ apply_fun g0 x0) x HxX) (from right to left) at position 1.
L182143
An exact proof term for the current goal is Himg.
L182144
An exact proof term for the current goal is (SepI X (λy : set ⇒ apply_fun g0 y ∈ V) x HxX Himg').
L182146
Let x be given.
L182146
Assume Hx: x ∈ preimage_of X g0 V.
L182146
We will prove x ∈ preimage_of X g0g V.
L182147
We prove the intermediate claim HxX: x ∈ X.
L182149
An exact proof term for the current goal is (SepE1 X (λy : set ⇒ apply_fun g0 y ∈ V) x Hx).
L182149
We prove the intermediate claim Himg: apply_fun g0 x ∈ V.
L182151
An exact proof term for the current goal is (SepE2 X (λy : set ⇒ apply_fun g0 y ∈ V) x Hx).
L182151
We prove the intermediate claim Himg': apply_fun g0g x ∈ V.
L182153
rewrite the current goal using (apply_fun_graph X (λx0 : set ⇒ apply_fun g0 x0) x HxX) (from left to right) at position 1.
L182153
An exact proof term for the current goal is Himg.
L182154
An exact proof term for the current goal is (SepI X (λy : set ⇒ apply_fun g0g y ∈ V) x HxX Himg').
L182155
rewrite the current goal using Heq (from left to right).
L182156
An exact proof term for the current goal is Hpre_g0.
L182157
We will prove continuous_map X Tx R R_standard_topology g0g.
L182158
We will prove topology_on X Tx ∧ topology_on R R_standard_topology ∧ function_on g0g X R ∧ ∀V : set, V ∈ R_standard_topology → preimage_of X g0g V ∈ Tx.
L182161
We prove the intermediate claim Htop: topology_on X Tx ∧ topology_on R R_standard_topology.
L182163
Apply andI to the current goal.
L182163
An exact proof term for the current goal is HTx'.
L182163
An exact proof term for the current goal is HTR.
L182163
We prove the intermediate claim Htopfun: (topology_on X Tx ∧ topology_on R R_standard_topology) ∧ function_on g0g X R.
L182165
Apply andI to the current goal.
L182165
An exact proof term for the current goal is Htop.
L182165
An exact proof term for the current goal is Hfun_g0g.
L182165
Apply andI to the current goal.
L182167
An exact proof term for the current goal is Htopfun.
L182168
An exact proof term for the current goal is Hpre_g0g.
L182168
We prove the intermediate claim HInv_nat: ∀n : set, nat_p n → continuous_map X Tx R R_standard_topology ((State n) 0).
L182170
Apply nat_ind to the current goal.
L182171
We will prove continuous_map X Tx R R_standard_topology ((State 0) 0).
L182171
We prove the intermediate claim H0: State 0 = BaseState.
L182173
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L182173
rewrite the current goal using H0 (from left to right).
L182174
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L182176
rewrite the current goal using HBase (from left to right).
L182177
rewrite the current goal using (tuple_2_0_eq g0g (f1s,den)) (from left to right).
L182178
An exact proof term for the current goal is Hg0g_cont.
L182180
Let n be given.
L182180
Assume HnNat: nat_p n.
L182180
Assume IH: continuous_map X Tx R R_standard_topology ((State n) 0).
L182181
We will prove continuous_map X Tx R R_standard_topology ((State (ordsucc n)) 0).
L182182
We prove the intermediate claim HS: State (ordsucc n) = StepState n (State n).
L182184
An exact proof term for the current goal is (nat_primrec_S BaseState StepState n HnNat).
L182184
rewrite the current goal using HS (from left to right).
L182185
Set st to be the term State n.
L182186
Set g to be the term st 0.
L182187
Set rpack to be the term st 1.
L182188
Set r to be the term rpack 0.
L182189
Set c to be the term rpack 1.
L182190
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L182193
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L182198
Set cNew to be the term mul_SNo c den.
L182199
We prove the intermediate claim HStep0: (StepState n st) 0 = gNew.
L182201
We prove the intermediate claim Hdef: StepState n st = (gNew,(rNew,cNew)).
Use reflexivity.
L182202
rewrite the current goal using Hdef (from left to right).
L182203
An exact proof term for the current goal is (tuple_2_0_eq gNew (rNew,cNew)).
L182204
rewrite the current goal using HStep0 (from left to right).
L182205
We prove the intermediate claim Hg_cont: continuous_map X Tx R R_standard_topology g.
L182207
An exact proof term for the current goal is IH.
L182207
We prove the intermediate claim HnO: n ∈ ω.
L182209
An exact proof term for the current goal is (nat_p_omega n HnNat).
L182209
We prove the intermediate claim Hcpack: ((State n) 1) 1 ∈ R ∧ 0 < ((State n) 1) 1.
L182211
An exact proof term for the current goal is (HInv_cpos n HnO).
L182211
We prove the intermediate claim Hc_eq: c = ((State n) 1) 1.
Use reflexivity.
L182213
We prove the intermediate claim HcR0: ((State n) 1) 1 ∈ R.
L182215
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) Hcpack).
L182215
We prove the intermediate claim Hcpos0: 0 < ((State n) 1) 1.
L182217
An exact proof term for the current goal is (andER (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) Hcpack).
L182217
We prove the intermediate claim HcR: c ∈ R.
L182219
rewrite the current goal using Hc_eq (from left to right).
L182219
An exact proof term for the current goal is HcR0.
L182219
We prove the intermediate claim Hcpos: 0 < c.
L182221
rewrite the current goal using Hc_eq (from left to right).
L182221
An exact proof term for the current goal is Hcpos0.
L182221
We prove the intermediate claim Hr_contI: continuous_map A Ta I Ti r.
L182223
An exact proof term for the current goal is (HInv_r_contI n HnO).
L182223
We prove the intermediate claim Hu_pack: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x = one_third).
L182230
An exact proof term for the current goal is (Hu_of_prop r Hr_contI).
L182230
We prove the intermediate claim Hu_left: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third).
L182235
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x = one_third) Hu_pack).
L182241
We prove the intermediate claim Hu_cont: continuous_map X Tx I0 T0 (u_of r).
L182243
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r)) (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third) Hu_left).
L182247
We prove the intermediate claim HI0SubR: I0 ⊆ R.
L182249
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L182249
We prove the intermediate claim HT0: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L182251
We prove the intermediate claim Hmul_contR: continuous_map R R_standard_topology R R_standard_topology (mul_const_fun c).
L182253
An exact proof term for the current goal is (mul_const_fun_continuous_pos c HcR Hcpos).
L182253
We prove the intermediate claim Hmul_contI0: continuous_map I0 T0 R R_standard_topology (mul_const_fun c).
L182255
rewrite the current goal using HT0 (from left to right).
L182255
An exact proof term for the current goal is (continuous_on_subspace R R_standard_topology R R_standard_topology (mul_const_fun c) I0 R_standard_topology_is_topology HI0SubR Hmul_contR).
L182257
We prove the intermediate claim Hh_cont: continuous_map X Tx R R_standard_topology (compose_fun X (u_of r) (mul_const_fun c)).
L182259
An exact proof term for the current goal is (composition_continuous X Tx I0 T0 R R_standard_topology (u_of r) (mul_const_fun c) Hu_cont Hmul_contI0).
L182260
We prove the intermediate claim Hpair_cont: continuous_map X Tx (setprod R R) (product_topology R R_standard_topology R R_standard_topology) (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))).
L182264
An exact proof term for the current goal is (maps_into_products_axiom X Tx R R_standard_topology R R_standard_topology g (compose_fun X (u_of r) (mul_const_fun c)) Hg_cont Hh_cont).
L182266
We prove the intermediate claim Hadd_cont: continuous_map (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology add_fun_R.
L182270
An exact proof term for the current goal is add_fun_R_continuous.
L182270
An exact proof term for the current goal is (composition_continuous X Tx (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R Hpair_cont Hadd_cont).
L182277
Let n be given.
L182278
Assume HnO: n ∈ ω.
L182278
An exact proof term for the current goal is (HInv_nat n (omega_nat_p n HnO)).
L182279
Let n be given.
L182280
Assume HnO: n ∈ ω.
L182280
We will prove continuous_map X Tx R R_standard_topology (apply_fun fn n).
L182281
We prove the intermediate claim Hdef: fn = graph ω (λn0 : set ⇒ (State n0) 0).
Use reflexivity.
L182283
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λn0 : set ⇒ (State n0) 0) n Hdef HnO) (from left to right).
L182284
An exact proof term for the current goal is (HInv_g_cont n HnO).
L182286
We prove the intermediate claim HInv_g_I: ∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun ((State n) 0) x ∈ I.
(*** range-in-I of each fn n (invariant on State n) ***)
L182288
We prove the intermediate claim HInv_nat: ∀n : set, nat_p n → ∀x : set, x ∈ X → apply_fun ((State n) 0) x ∈ I ∧ apply_fun ((State n) 0) x ∈ closed_interval (add_SNo (minus_SNo 1) (((State n) 1) 1)) (add_SNo 1 (minus_SNo (((State n) 1) 1))).
L182292
Apply nat_ind to the current goal.
L182293
Let x be given.
L182293
Assume HxX: x ∈ X.
L182293
We will prove apply_fun ((State 0) 0) x ∈ I ∧ apply_fun ((State 0) 0) x ∈ closed_interval (add_SNo (minus_SNo 1) (((State 0) 1) 1)) (add_SNo 1 (minus_SNo (((State 0) 1) 1))).
L182296
We prove the intermediate claim H0: State 0 = BaseState.
L182298
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L182298
rewrite the current goal using H0 (from left to right).
L182299
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L182301
rewrite the current goal using HBase (from left to right).
L182302
rewrite the current goal using (tuple_2_0_eq g0g (f1s,den)) (from left to right).
L182303
rewrite the current goal using (tuple_2_0_eq g0g (f1s,den)) (from left to right).
L182304
We prove the intermediate claim Hc0Eq: ((g0g,(f1s,den)) 1) 1 = den.
L182306
rewrite the current goal using (tuple_2_1_eq g0g (f1s,den)) (from left to right) at position 1.
L182306
rewrite the current goal using (tuple_2_1_eq f1s den) (from left to right) at position 1.
Use reflexivity.
L182308
Apply andI to the current goal.
L182310
rewrite the current goal using (apply_fun_graph X (λx0 : set ⇒ apply_fun g0 x0) x HxX) (from left to right).
L182311
We prove the intermediate claim Hg0xI0: apply_fun g0 x ∈ I0.
(*** in I ***)
L182313
An exact proof term for the current goal is (Hfung0 x HxX).
L182313
An exact proof term for the current goal is (closed_interval_minus_one_third_one_third_sub_closed_interval_minus1_1 (apply_fun g0 x) Hg0xI0).
L182315
rewrite the current goal using (apply_fun_graph X (λx0 : set ⇒ apply_fun g0 x0) x HxX) (from left to right).
L182316
We prove the intermediate claim Hg0xI0: apply_fun g0 x ∈ I0.
(*** in the shrinking interval for c0=den=2/3 ***)
L182318
An exact proof term for the current goal is (Hfung0 x HxX).
L182318
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
L182320
We prove the intermediate claim HlowEq0: add_SNo (minus_SNo 1) (((g0g,(f1s,den)) 1) 1) = minus_SNo one_third.
L182323
rewrite the current goal using Hc0Eq (from left to right).
L182323
rewrite the current goal using HdenDef (from left to right).
L182324
An exact proof term for the current goal is add_minus1_two_thirds_eq_minus_one_third.
L182325
We prove the intermediate claim HupEq0: add_SNo 1 (minus_SNo (((g0g,(f1s,den)) 1) 1)) = one_third.
L182328
rewrite the current goal using Hc0Eq (from left to right).
L182328
rewrite the current goal using HdenDef (from left to right).
L182329
An exact proof term for the current goal is add_1_minus_two_thirds_eq_one_third.
L182330
rewrite the current goal using HlowEq0 (from left to right).
L182331
rewrite the current goal using HupEq0 (from left to right).
L182332
An exact proof term for the current goal is Hg0xI0.
L182334
Let n be given.
L182334
Assume HnNat: nat_p n.
L182334
Assume IH: ∀x : set, x ∈ X → apply_fun ((State n) 0) x ∈ I ∧ apply_fun ((State n) 0) x ∈ closed_interval (add_SNo (minus_SNo 1) (((State n) 1) 1)) (add_SNo 1 (minus_SNo (((State n) 1) 1))).
L182338
Let x be given.
L182339
Assume HxX: x ∈ X.
L182339
We will prove apply_fun ((State (ordsucc n)) 0) x ∈ I ∧ apply_fun ((State (ordsucc n)) 0) x ∈ closed_interval (add_SNo (minus_SNo 1) (((State (ordsucc n)) 1) 1)) (add_SNo 1 (minus_SNo (((State (ordsucc n)) 1) 1))).
L182342
We prove the intermediate claim HSnat: State (ordsucc n) = StepState n (State n).
(*** TODO: step case needs the geometric-series bound plus range control of the correction term. ***)
L182345
An exact proof term for the current goal is (nat_primrec_S BaseState StepState n HnNat).
L182345
rewrite the current goal using HSnat (from left to right).
L182346
Set st to be the term State n.
L182347
Set g to be the term st 0.
L182348
Set rpack to be the term st 1.
L182349
Set r to be the term rpack 0.
L182350
Set c to be the term rpack 1.
L182351
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L182354
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L182359
Set cNew to be the term mul_SNo c den.
L182360
We prove the intermediate claim HdefStep: StepState n st = (gNew,(rNew,cNew)).
Use reflexivity.
L182362
rewrite the current goal using HdefStep (from left to right).
L182363
We prove the intermediate claim HgStep: ((gNew,(rNew,cNew)) 0) = gNew.
L182365
An exact proof term for the current goal is (tuple_2_0_eq gNew (rNew,cNew)).
L182365
rewrite the current goal using HgStep (from left to right).
L182366
We prove the intermediate claim HnO: n ∈ ω.
L182368
An exact proof term for the current goal is (nat_p_omega n HnNat).
L182368
We prove the intermediate claim HrEq: r = ((State n) 1) 0.
Use reflexivity.
L182370
We prove the intermediate claim HcEq: c = ((State n) 1) 1.
Use reflexivity.
L182372
We prove the intermediate claim Hr_contI: continuous_map A Ta I Ti r.
L182374
rewrite the current goal using HrEq (from left to right).
L182374
An exact proof term for the current goal is (HInv_r_contI n HnO).
L182375
We prove the intermediate claim Hu_pack: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x0 = one_third).
L182382
An exact proof term for the current goal is (Hu_of_prop r Hr_contI).
L182382
We prove the intermediate claim Hu_contI0: continuous_map X Tx I0 T0 (u_of r).
L182384
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r)) (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third) (andEL ((continuous_map X Tx I0 T0 (u_of r)) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x0 = one_third) Hu_pack)).
L182394
We prove the intermediate claim Hu_fun: function_on (u_of r) X I0.
L182396
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 (u_of r) Hu_contI0).
L182396
We prove the intermediate claim HuxI0: apply_fun (u_of r) x ∈ I0.
L182398
An exact proof term for the current goal is (Hu_fun x HxX).
L182398
We prove the intermediate claim HuxR: apply_fun (u_of r) x ∈ R.
L182400
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun (u_of r) x) HuxI0).
L182400
We prove the intermediate claim HcR: c ∈ R.
L182402
rewrite the current goal using HcEq (from left to right).
L182402
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L182406
We prove the intermediate claim HgEq: g = (State n) 0.
Use reflexivity.
L182408
We prove the intermediate claim HgxI: apply_fun g x ∈ I.
L182410
rewrite the current goal using HgEq (from left to right).
L182410
An exact proof term for the current goal is (andEL (apply_fun ((State n) 0) x ∈ I) (apply_fun ((State n) 0) x ∈ closed_interval (add_SNo (minus_SNo 1) (((State n) 1) 1)) (add_SNo 1 (minus_SNo (((State n) 1) 1)))) (IH x HxX)).
L182415
We prove the intermediate claim HgxR: apply_fun g x ∈ R.
L182417
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun g x) HgxI).
L182417
We prove the intermediate claim HcorrEq: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x = mul_SNo (apply_fun (u_of r) x) c.
L182421
We prove the intermediate claim Hcomp: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x = apply_fun (mul_const_fun c) (apply_fun (u_of r) x).
L182424
An exact proof term for the current goal is (compose_fun_apply X (u_of r) (mul_const_fun c) x HxX).
L182424
rewrite the current goal using Hcomp (from left to right).
L182425
An exact proof term for the current goal is (mul_const_fun_apply c (apply_fun (u_of r) x) HcR HuxR).
L182426
We prove the intermediate claim HcorrR: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x ∈ R.
L182428
rewrite the current goal using HcorrEq (from left to right).
L182428
An exact proof term for the current goal is (real_mul_SNo (apply_fun (u_of r) x) HuxR c HcR).
L182429
We prove the intermediate claim HgNewEval: apply_fun gNew x = add_SNo (apply_fun g x) (apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x).
L182433
An exact proof term for the current goal is (add_of_pair_map_apply X g (compose_fun X (u_of r) (mul_const_fun c)) x HxX HgxR HcorrR).
L182433
rewrite the current goal using HgNewEval (from left to right).
L182435
rewrite the current goal using HcorrEq (from left to right).
(*** TODO: finish by bounding the correction term and using interval closure under this controlled addition. ***)
L182436
We prove the intermediate claim Hcpos: 0 < c.
L182438
rewrite the current goal using HcEq (from left to right).
L182438
An exact proof term for the current goal is (andER (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L182442
We prove the intermediate claim HyIshrink: apply_fun g x ∈ closed_interval (add_SNo (minus_SNo 1) c) (add_SNo 1 (minus_SNo c)).
L182445
rewrite the current goal using HgEq (from left to right).
L182445
rewrite the current goal using HcEq (from left to right).
L182446
An exact proof term for the current goal is (andER (apply_fun ((State n) 0) x ∈ I) (apply_fun ((State n) 0) x ∈ closed_interval (add_SNo (minus_SNo 1) (((State n) 1) 1)) (add_SNo 1 (minus_SNo (((State n) 1) 1)))) (IH x HxX)).
L182451
We prove the intermediate claim HzIthird: mul_SNo (apply_fun (u_of r) x) c ∈ closed_interval (minus_SNo (mul_SNo c one_third)) (mul_SNo c one_third).
L182455
We will prove mul_SNo (apply_fun (u_of r) x) c ∈ closed_interval (minus_SNo (mul_SNo c one_third)) (mul_SNo c one_third).
L182455
We prove the intermediate claim H13R: one_third ∈ R.
L182457
An exact proof term for the current goal is one_third_in_R.
L182457
We prove the intermediate claim H13S: SNo one_third.
L182459
An exact proof term for the current goal is (real_SNo one_third H13R).
L182459
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L182461
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L182461
We prove the intermediate claim Hm13S: SNo (minus_SNo one_third).
L182463
An exact proof term for the current goal is (SNo_minus_SNo one_third H13S).
L182463
We prove the intermediate claim HuS: SNo (apply_fun (u_of r) x).
L182465
An exact proof term for the current goal is (real_SNo (apply_fun (u_of r) x) HuxR).
L182465
We prove the intermediate claim HcS: SNo c.
L182467
An exact proof term for the current goal is (real_SNo c HcR).
L182467
We prove the intermediate claim Hbnds0: Rle (minus_SNo one_third) (apply_fun (u_of r) x) ∧ Rle (apply_fun (u_of r) x) one_third.
L182471
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun (u_of r) x) Hm13R H13R HuxI0).
L182472
We prove the intermediate claim HloRle: Rle (minus_SNo one_third) (apply_fun (u_of r) x).
L182474
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun (u_of r) x)) (Rle (apply_fun (u_of r) x) one_third) Hbnds0).
L182477
We prove the intermediate claim HhiRle: Rle (apply_fun (u_of r) x) one_third.
L182479
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun (u_of r) x)) (Rle (apply_fun (u_of r) x) one_third) Hbnds0).
L182482
We prove the intermediate claim Hnot_lt_lo: ¬ ((apply_fun (u_of r) x) < (minus_SNo one_third)).
L182484
Assume Hlt: (apply_fun (u_of r) x) < (minus_SNo one_third).
L182484
We prove the intermediate claim Hrlt: Rlt (apply_fun (u_of r) x) (minus_SNo one_third).
L182486
An exact proof term for the current goal is (RltI (apply_fun (u_of r) x) (minus_SNo one_third) HuxR Hm13R Hlt).
L182486
An exact proof term for the current goal is ((RleE_nlt (minus_SNo one_third) (apply_fun (u_of r) x) HloRle) Hrlt).
L182487
We prove the intermediate claim Hnot_lt_hi: ¬ (one_third < (apply_fun (u_of r) x)).
L182489
Assume Hlt: one_third < (apply_fun (u_of r) x).
L182489
We prove the intermediate claim Hrlt: Rlt one_third (apply_fun (u_of r) x).
L182491
An exact proof term for the current goal is (RltI one_third (apply_fun (u_of r) x) H13R HuxR Hlt).
L182491
An exact proof term for the current goal is ((RleE_nlt (apply_fun (u_of r) x) one_third HhiRle) Hrlt).
L182492
We prove the intermediate claim Hm13_le_u: (minus_SNo one_third) ≤ (apply_fun (u_of r) x).
L182494
Apply (SNoLtLe_or (apply_fun (u_of r) x) (minus_SNo one_third) HuS Hm13S) to the current goal.
L182495
Assume Hlt: (apply_fun (u_of r) x) < (minus_SNo one_third).
L182495
An exact proof term for the current goal is (FalseE (Hnot_lt_lo Hlt) ((minus_SNo one_third) ≤ (apply_fun (u_of r) x))).
L182497
Assume Hle: (minus_SNo one_third) ≤ (apply_fun (u_of r) x).
L182497
An exact proof term for the current goal is Hle.
L182498
We prove the intermediate claim Hu_le_13: (apply_fun (u_of r) x) ≤ one_third.
L182500
Apply (SNoLtLe_or one_third (apply_fun (u_of r) x) H13S HuS) to the current goal.
L182501
Assume Hlt: one_third < (apply_fun (u_of r) x).
L182501
An exact proof term for the current goal is (FalseE (Hnot_lt_hi Hlt) ((apply_fun (u_of r) x) ≤ one_third)).
L182503
Assume Hle: (apply_fun (u_of r) x) ≤ one_third.
L182503
An exact proof term for the current goal is Hle.
L182504
We prove the intermediate claim H0le_c: 0 ≤ c.
L182506
An exact proof term for the current goal is (SNoLtLe 0 c Hcpos).
L182506
We prove the intermediate claim HmulLeHi: mul_SNo (apply_fun (u_of r) x) c ≤ mul_SNo one_third c.
L182508
An exact proof term for the current goal is (nonneg_mul_SNo_Le' (apply_fun (u_of r) x) one_third c HuS H13S HcS H0le_c Hu_le_13).
L182508
We prove the intermediate claim HmulLeLo: mul_SNo (minus_SNo one_third) c ≤ mul_SNo (apply_fun (u_of r) x) c.
L182510
An exact proof term for the current goal is (nonneg_mul_SNo_Le' (minus_SNo one_third) (apply_fun (u_of r) x) c Hm13S HuS HcS H0le_c Hm13_le_u).
L182510
We prove the intermediate claim HmulR: mul_SNo (apply_fun (u_of r) x) c ∈ R.
L182512
An exact proof term for the current goal is (real_mul_SNo (apply_fun (u_of r) x) HuxR c HcR).
L182512
We prove the intermediate claim Hc13R: mul_SNo c one_third ∈ R.
L182514
An exact proof term for the current goal is (real_mul_SNo c HcR one_third H13R).
L182514
We prove the intermediate claim Hm_c13R: minus_SNo (mul_SNo c one_third) ∈ R.
L182516
An exact proof term for the current goal is (real_minus_SNo (mul_SNo c one_third) Hc13R).
L182516
We prove the intermediate claim HhiEq: mul_SNo one_third c = mul_SNo c one_third.
L182518
An exact proof term for the current goal is (mul_SNo_com one_third c H13S HcS).
L182518
We prove the intermediate claim HloEq: mul_SNo (minus_SNo one_third) c = minus_SNo (mul_SNo c one_third).
L182520
rewrite the current goal using (mul_SNo_minus_distrL one_third c H13S HcS) (from left to right) at position 1.
L182520
rewrite the current goal using HhiEq (from left to right) at position 1.
Use reflexivity.
L182522
We prove the intermediate claim HhiLe': mul_SNo (apply_fun (u_of r) x) c ≤ mul_SNo c one_third.
L182524
rewrite the current goal using HhiEq (from right to left).
L182524
An exact proof term for the current goal is HmulLeHi.
L182525
We prove the intermediate claim HloLe': (minus_SNo (mul_SNo c one_third)) ≤ mul_SNo (apply_fun (u_of r) x) c.
L182527
rewrite the current goal using HloEq (from right to left) at position 1.
L182527
An exact proof term for the current goal is HmulLeLo.
L182528
We prove the intermediate claim HhiRle': Rle (mul_SNo (apply_fun (u_of r) x) c) (mul_SNo c one_third).
L182530
An exact proof term for the current goal is (Rle_of_SNoLe (mul_SNo (apply_fun (u_of r) x) c) (mul_SNo c one_third) HmulR Hc13R HhiLe').
L182530
We prove the intermediate claim HloRle': Rle (minus_SNo (mul_SNo c one_third)) (mul_SNo (apply_fun (u_of r) x) c).
L182532
An exact proof term for the current goal is (Rle_of_SNoLe (minus_SNo (mul_SNo c one_third)) (mul_SNo (apply_fun (u_of r) x) c) Hm_c13R HmulR HloLe').
L182532
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo (mul_SNo c one_third)) (mul_SNo c one_third) (mul_SNo (apply_fun (u_of r) x) c) Hm_c13R Hc13R HmulR HloRle' HhiRle').
L182536
We prove the intermediate claim HsumShrink: add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c) ∈ closed_interval (add_SNo (minus_SNo 1) (mul_SNo c two_thirds)) (add_SNo 1 (minus_SNo (mul_SNo c two_thirds))).
L182541
An exact proof term for the current goal is (add_SNo_interval_expand_by_third_stub c (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c) HcR Hcpos HyIshrink HzIthird).
L182544
Apply andI to the current goal.
L182546
Set t to be the term mul_SNo c two_thirds.
L182547
We prove the intermediate claim H23R: two_thirds ∈ R.
(*** in I (existing argument via HsumShrink) ***)
L182549
An exact proof term for the current goal is two_thirds_in_R.
L182549
We prove the intermediate claim HtR: t ∈ R.
L182551
An exact proof term for the current goal is (real_mul_SNo c HcR two_thirds H23R).
L182551
We prove the intermediate claim Htpos: 0 < t.
L182553
We prove the intermediate claim H23S: SNo two_thirds.
L182554
An exact proof term for the current goal is (real_SNo two_thirds H23R).
L182554
We prove the intermediate claim HcS: SNo c.
L182556
An exact proof term for the current goal is (real_SNo c HcR).
L182556
An exact proof term for the current goal is (mul_SNo_pos_pos c two_thirds HcS H23S Hcpos two_thirds_pos).
L182557
We prove the intermediate claim H0lt_t: Rlt 0 t.
L182559
An exact proof term for the current goal is (RltI 0 t real_0 HtR Htpos).
L182559
We prove the intermediate claim H0le_t: Rle 0 t.
L182561
An exact proof term for the current goal is (Rlt_implies_Rle 0 t H0lt_t).
L182561
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L182563
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L182563
We prove the intermediate claim Hm1S: SNo (minus_SNo 1).
L182565
An exact proof term for the current goal is (SNo_minus_SNo 1 SNo_1).
L182565
We prove the intermediate claim Ha_low: Rle (minus_SNo 1) (add_SNo (minus_SNo 1) t).
L182567
We prove the intermediate claim Htmp: Rle (add_SNo (minus_SNo 1) 0) (add_SNo (minus_SNo 1) t).
L182568
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo 1) 0 t Hm1R real_0 HtR H0le_t).
L182568
rewrite the current goal using (add_SNo_0R (minus_SNo 1) Hm1S) (from right to left) at position 1.
L182569
An exact proof term for the current goal is Htmp.
L182570
We prove the intermediate claim Hneg_t_le0: Rle (minus_SNo t) 0.
L182572
An exact proof term for the current goal is (Rle_minus_nonneg t HtR (RleE_nlt 0 t H0le_t)).
L182572
We prove the intermediate claim Hb_up: Rle (add_SNo 1 (minus_SNo t)) 1.
L182574
We prove the intermediate claim Htmp: Rle (add_SNo 1 (minus_SNo t)) (add_SNo 1 0).
L182575
An exact proof term for the current goal is (Rle_add_SNo_2 1 (minus_SNo t) 0 real_1 (real_minus_SNo t HtR) real_0 Hneg_t_le0).
L182575
We prove the intermediate claim H10Eq: add_SNo 1 0 = 1.
L182577
An exact proof term for the current goal is (add_SNo_0R 1 SNo_1).
L182577
We prove the intermediate claim Hmid: Rle (add_SNo 1 0) 1.
L182579
rewrite the current goal using H10Eq (from left to right) at position 1.
L182579
An exact proof term for the current goal is (Rle_refl 1 real_1).
L182580
An exact proof term for the current goal is (Rle_tra (add_SNo 1 (minus_SNo t)) (add_SNo 1 0) 1 Htmp Hmid).
L182581
We prove the intermediate claim HyR: add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c) ∈ R.
L182584
An exact proof term for the current goal is (closed_interval_sub_R (add_SNo (minus_SNo 1) (mul_SNo c two_thirds)) (add_SNo 1 (minus_SNo (mul_SNo c two_thirds))) (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) HsumShrink).
L182587
We prove the intermediate claim Hbnds: Rle (add_SNo (minus_SNo 1) t) (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) ∧ Rle (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) (add_SNo 1 (minus_SNo t)).
L182591
An exact proof term for the current goal is (closed_interval_bounds (add_SNo (minus_SNo 1) t) (add_SNo 1 (minus_SNo t)) (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) (real_add_SNo (minus_SNo 1) Hm1R t HtR) (real_add_SNo 1 real_1 (minus_SNo t) (real_minus_SNo t HtR)) HsumShrink).
L182595
We prove the intermediate claim Hlo: Rle (minus_SNo 1) (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)).
L182598
An exact proof term for the current goal is (Rle_tra (minus_SNo 1) (add_SNo (minus_SNo 1) t) (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) Ha_low (andEL (Rle (add_SNo (minus_SNo 1) t) (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c))) (Rle (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) (add_SNo 1 (minus_SNo t))) Hbnds)).
L182604
We prove the intermediate claim Hhi: Rle (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) 1.
L182607
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) (add_SNo 1 (minus_SNo t)) 1 (andER (Rle (add_SNo (minus_SNo 1) t) (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c))) (Rle (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) (add_SNo 1 (minus_SNo t))) Hbnds) Hb_up).
L182613
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) Hm1R real_1 HyR Hlo Hhi).
L182617
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
(*** in the shrinking interval for the successor scalar ***)
L182619
We prove the intermediate claim HcNewEq: cNew = mul_SNo c two_thirds.
L182621
rewrite the current goal using HdenDef (from left to right).
Use reflexivity.
L182622
We prove the intermediate claim HcPairEq: ((gNew,(rNew,cNew)) 1) 1 = cNew.
L182624
We prove the intermediate claim Hinner: ((gNew,(rNew,cNew)) 1) = (rNew,cNew).
L182625
An exact proof term for the current goal is (tuple_2_1_eq gNew (rNew,cNew)).
L182625
rewrite the current goal using Hinner (from left to right) at position 1.
L182626
An exact proof term for the current goal is (tuple_2_1_eq rNew cNew).
L182627
We prove the intermediate claim HcGoalEq: ((gNew,(rNew,cNew)) 1) 1 = mul_SNo c two_thirds.
L182629
rewrite the current goal using HcPairEq (from left to right).
L182629
rewrite the current goal using HcNewEq (from left to right).
Use reflexivity.
L182631
We prove the intermediate claim HlowEq: add_SNo (minus_SNo 1) (((gNew,(rNew,cNew)) 1) 1) = add_SNo (minus_SNo 1) (mul_SNo c two_thirds).
L182635
rewrite the current goal using HcGoalEq (from left to right).
Use reflexivity.
L182636
We prove the intermediate claim HupEq: add_SNo 1 (minus_SNo (((gNew,(rNew,cNew)) 1) 1)) = add_SNo 1 (minus_SNo (mul_SNo c two_thirds)).
L182640
rewrite the current goal using HcGoalEq (from left to right).
Use reflexivity.
L182641
rewrite the current goal using HlowEq (from left to right).
L182642
rewrite the current goal using HupEq (from left to right).
L182643
An exact proof term for the current goal is HsumShrink.
L182644
Let n be given.
L182645
Assume HnO: n ∈ ω.
L182645
Let x be given.
L182646
Assume HxX: x ∈ X.
L182646
An exact proof term for the current goal is (andEL (apply_fun ((State n) 0) x ∈ I) (apply_fun ((State n) 0) x ∈ closed_interval (add_SNo (minus_SNo 1) (((State n) 1) 1)) (add_SNo 1 (minus_SNo (((State n) 1) 1)))) (HInv_nat n (omega_nat_p n HnO) x HxX)).
L182651
Let n be given.
L182652
Assume HnO: n ∈ ω.
L182652
Let x be given.
L182653
Assume HxX: x ∈ X.
L182653
We will prove apply_fun (apply_fun fn n) x ∈ I.
L182654
We prove the intermediate claim Hdef: fn = graph ω (λn0 : set ⇒ (State n0) 0).
Use reflexivity.
L182656
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λn0 : set ⇒ (State n0) 0) n Hdef HnO) (from left to right).
L182657
An exact proof term for the current goal is (HInv_g_I n HnO x HxX).
L182659
Let x be given.
L182660
Assume HxA: x ∈ A.
L182660
We will prove converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x).
(*** pointwise convergence on A (to be proved from residual convergence / geometric tail bounds) ***)
L182663
Set seq1 to be the term graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x).
L182664
Set seq2 to be the term graph ω (λn : set ⇒ apply_fun ((State n) 0) x).
L182665
We prove the intermediate claim HseqEq: seq1 = seq2.
L182667
Apply set_ext to the current goal.
L182668
Let p be given.
L182668
Assume Hp: p ∈ seq1.
L182668
We will prove p ∈ seq2.
L182669
Apply (ReplE_impred ω (λn0 : set ⇒ (n0,apply_fun (apply_fun fn n0) x)) p Hp (p ∈ seq2)) to the current goal.
L182670
Let n be given.
L182671
Assume HnO: n ∈ ω.
L182671
Assume HpEq: p = (n,apply_fun (apply_fun fn n) x).
L182672
rewrite the current goal using HpEq (from left to right).
L182673
We prove the intermediate claim Hdef: fn = graph ω (λn0 : set ⇒ (State n0) 0).
Use reflexivity.
L182675
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λn0 : set ⇒ (State n0) 0) n Hdef HnO) (from left to right) at position 1.
L182676
An exact proof term for the current goal is (ReplI ω (λn0 : set ⇒ (n0,apply_fun ((State n0) 0) x)) n HnO).
L182678
Let p be given.
L182678
Assume Hp: p ∈ seq2.
L182678
We will prove p ∈ seq1.
L182679
Apply (ReplE_impred ω (λn0 : set ⇒ (n0,apply_fun ((State n0) 0) x)) p Hp (p ∈ seq1)) to the current goal.
L182680
Let n be given.
L182681
Assume HnO: n ∈ ω.
L182681
Assume HpEq: p = (n,apply_fun ((State n) 0) x).
L182682
rewrite the current goal using HpEq (from left to right).
L182683
We prove the intermediate claim Hdef: fn = graph ω (λn0 : set ⇒ (State n0) 0).
Use reflexivity.
L182685
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λn0 : set ⇒ (State n0) 0) n Hdef HnO) (from right to left) at position 1.
L182686
An exact proof term for the current goal is (ReplI ω (λn0 : set ⇒ (n0,apply_fun (apply_fun fn n0) x)) n HnO).
L182687
We prove the intermediate claim Hseq1def: seq1 = (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)).
Use reflexivity.
L182690
rewrite the current goal using Hseq1def (from right to left).
L182691
rewrite the current goal using HseqEq (from left to right).
L182692
We prove the intermediate claim Hseq2_conv: converges_to R (metric_topology R R_bounded_metric) seq2 (apply_fun f x).
(*** TODO: show seq2 converges to f x in the bounded-metric topology using the residual recursion. ***)
L182698
Set lim to be the term apply_fun f x.
L182699
We prove the intermediate claim HconvM: sequence_converges_metric R R_bounded_metric seq2 lim.
(*** TODO: derive the usual bound |(State n)0 x - f x| <= ((State n)1)1 on A and use ((State n)1)1 -> 0. ***)
L182701
We will prove metric_on R R_bounded_metric ∧ sequence_on seq2 R ∧ lim ∈ R ∧ ∀eps : set, eps ∈ R ∧ Rlt 0 eps → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → Rlt (apply_fun R_bounded_metric (apply_fun seq2 n,lim)) eps.
L182705
We prove the intermediate claim Hseq2On: sequence_on seq2 R.
L182707
We will prove sequence_on seq2 R.
L182707
Let n be given.
L182708
Assume HnO: n ∈ ω.
L182708
We will prove apply_fun seq2 n ∈ R.
L182709
We prove the intermediate claim HxX: x ∈ X.
L182711
An exact proof term for the current goal is (HAsubX x HxA).
L182711
We prove the intermediate claim Hseq2def: seq2 = graph ω (λn0 : set ⇒ apply_fun ((State n0) 0) x).
Use reflexivity.
L182713
rewrite the current goal using Hseq2def (from left to right).
L182714
rewrite the current goal using (apply_fun_graph ω (λn0 : set ⇒ apply_fun ((State n0) 0) x) n HnO) (from left to right).
L182715
We prove the intermediate claim HgR_nat: ∀k : set, nat_p k → apply_fun ((State k) 0) x ∈ R.
L182717
Apply nat_ind to the current goal.
L182718
We will prove apply_fun ((State 0) 0) x ∈ R.
L182718
We prove the intermediate claim H0: State 0 = BaseState.
L182720
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L182720
rewrite the current goal using H0 (from left to right).
L182721
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L182723
rewrite the current goal using HBase (from left to right).
L182724
rewrite the current goal using (tuple_2_0_eq g0g (f1s,den)) (from left to right).
L182725
We prove the intermediate claim HxX0: x ∈ X.
L182727
An exact proof term for the current goal is (HAsubX x HxA).
L182727
rewrite the current goal using (apply_fun_graph X (λx0 : set ⇒ apply_fun g0 x0) x HxX0) (from left to right).
L182728
We prove the intermediate claim Hg0I0: apply_fun g0 x ∈ I0.
L182730
An exact proof term for the current goal is (Hfung0 x HxX0).
L182730
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun g0 x) Hg0I0).
L182732
Let k be given.
L182732
Assume HkNat: nat_p k.
L182732
Assume IHk: apply_fun ((State k) 0) x ∈ R.
L182733
We will prove apply_fun ((State (ordsucc k)) 0) x ∈ R.
L182734
We prove the intermediate claim HS: State (ordsucc k) = StepState k (State k).
L182736
An exact proof term for the current goal is (nat_primrec_S BaseState StepState k HkNat).
L182736
rewrite the current goal using HS (from left to right).
L182737
Set st to be the term State k.
L182738
Set g to be the term st 0.
L182739
Set rpack to be the term st 1.
L182740
Set r to be the term rpack 0.
L182741
Set c to be the term rpack 1.
L182742
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L182745
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L182750
Set cNew to be the term mul_SNo c den.
L182751
We prove the intermediate claim Hdef: StepState k st = (gNew,(rNew,cNew)).
Use reflexivity.
L182753
We prove the intermediate claim HgStep: (StepState k st) 0 = gNew.
L182755
rewrite the current goal using Hdef (from left to right).
L182755
An exact proof term for the current goal is (tuple_2_0_eq gNew (rNew,cNew)).
L182756
rewrite the current goal using HgStep (from left to right).
L182757
We prove the intermediate claim HxX0: x ∈ X.
L182759
An exact proof term for the current goal is (HAsubX x HxA).
L182759
We prove the intermediate claim HgEq: g = (State k) 0.
Use reflexivity.
L182761
We prove the intermediate claim HgxR: apply_fun g x ∈ R.
L182763
rewrite the current goal using HgEq (from left to right).
L182763
An exact proof term for the current goal is IHk.
L182764
We prove the intermediate claim HkO: k ∈ ω.
L182766
An exact proof term for the current goal is (nat_p_omega k HkNat).
L182766
We prove the intermediate claim HrEq: r = ((State k) 1) 0.
Use reflexivity.
L182768
We prove the intermediate claim Hr_contI: continuous_map A Ta I Ti r.
L182770
rewrite the current goal using HrEq (from left to right).
L182770
An exact proof term for the current goal is (HInv_r_contI k HkO).
L182771
We prove the intermediate claim Hu_pack: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x0 = one_third).
L182778
An exact proof term for the current goal is (Hu_of_prop r Hr_contI).
L182778
We prove the intermediate claim Hu_contI0: continuous_map X Tx I0 T0 (u_of r).
L182780
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r)) (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third) (andEL ((continuous_map X Tx I0 T0 (u_of r)) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x0 = one_third) Hu_pack)).
L182790
We prove the intermediate claim Hu_fun: function_on (u_of r) X I0.
L182792
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 (u_of r) Hu_contI0).
L182792
We prove the intermediate claim HuxI0: apply_fun (u_of r) x ∈ I0.
L182794
An exact proof term for the current goal is (Hu_fun x HxX0).
L182794
We prove the intermediate claim HuxR: apply_fun (u_of r) x ∈ R.
L182796
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun (u_of r) x) HuxI0).
L182796
We prove the intermediate claim HcEq: c = ((State k) 1) 1.
Use reflexivity.
L182798
We prove the intermediate claim HcR: c ∈ R.
L182800
rewrite the current goal using HcEq (from left to right).
L182800
An exact proof term for the current goal is (andEL (((State k) 1) 1 ∈ R) (0 < ((State k) 1) 1) (HInv_cpos k HkO)).
L182804
We prove the intermediate claim HcorrEq: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x = mul_SNo (apply_fun (u_of r) x) c.
L182808
We prove the intermediate claim Hcomp: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x = apply_fun (mul_const_fun c) (apply_fun (u_of r) x).
L182811
An exact proof term for the current goal is (compose_fun_apply X (u_of r) (mul_const_fun c) x HxX0).
L182811
rewrite the current goal using Hcomp (from left to right).
L182812
An exact proof term for the current goal is (mul_const_fun_apply c (apply_fun (u_of r) x) HcR HuxR).
L182813
We prove the intermediate claim HcorrR: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x ∈ R.
L182815
rewrite the current goal using HcorrEq (from left to right).
L182815
An exact proof term for the current goal is (real_mul_SNo (apply_fun (u_of r) x) HuxR c HcR).
L182816
We prove the intermediate claim HgNewEval: apply_fun gNew x = add_SNo (apply_fun g x) (apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x).
L182820
An exact proof term for the current goal is (add_of_pair_map_apply X g (compose_fun X (u_of r) (mul_const_fun c)) x HxX0 HgxR HcorrR).
L182820
rewrite the current goal using HgNewEval (from left to right).
L182821
An exact proof term for the current goal is (real_add_SNo (apply_fun g x) HgxR (apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x) HcorrR).
L182824
An exact proof term for the current goal is (HgR_nat n (omega_nat_p n HnO)).
L182825
Apply andI to the current goal.
L182827
We will prove (metric_on R R_bounded_metric ∧ sequence_on seq2 R) ∧ lim ∈ R.
L182827
Apply andI to the current goal.
L182829
We will prove metric_on R R_bounded_metric ∧ sequence_on seq2 R.
L182829
Apply andI to the current goal.
L182831
An exact proof term for the current goal is R_bounded_metric_is_metric_on_early.
L182832
An exact proof term for the current goal is Hseq2On.
L182833
We prove the intermediate claim HlimR: lim ∈ R.
L182834
An exact proof term for the current goal is (Hf_R x HxA).
L182834
An exact proof term for the current goal is HlimR.
L182836
Let eps be given.
L182836
Assume Heps: eps ∈ R ∧ Rlt 0 eps.
L182836
We prove the intermediate claim HepsR: eps ∈ R.
(*** ABY: try_aby at line 132949 column 1 failed ***)
(*** TODO: use the Step II residual invariant on A to get abs((State n)0 x - f x) <= ((State n)1)1,
then pick N so that ((State n)1)1 < min(eps,one_third) and conclude using
abs_lt_lt1_imp_R_bounded_distance_lt. ***)
L182842
An exact proof term for the current goal is (andEL (eps ∈ R) (Rlt 0 eps) Heps).
L182842
We prove the intermediate claim HepsPos: Rlt 0 eps.
L182844
An exact proof term for the current goal is (andER (eps ∈ R) (Rlt 0 eps) Heps).
L182844
We prove the intermediate claim HepsS: SNo eps.
L182846
An exact proof term for the current goal is (real_SNo eps HepsR).
L182846
Apply (SNoLt_trichotomy_or_impred eps 1 HepsS SNo_1 (∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → Rlt (apply_fun R_bounded_metric (apply_fun seq2 n,lim)) eps)) to the current goal.
L182851
Assume HepsLt1S: eps < 1.
L182851
We prove the intermediate claim HepsLt1: Rlt eps 1.
(*** TODO: show eps-step for eps<1 using the residual bound on A and a decay bound for ((State n)1)1. ***)
L182854
An exact proof term for the current goal is (RltI eps 1 HepsR real_1 HepsLt1S).
L182854
We prove the intermediate claim Hex_c_small: ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → ((State n) 1) 1 < eps.
L182859
We prove the intermediate claim HexK: ∃K ∈ ω, eps_ K < eps.
(*** reduce to a dyadic bound eps_ K and a geometric bound for the scalar component ***)
L182861
An exact proof term for the current goal is (exists_eps_lt_pos_Euclid eps HepsR HepsPos).
L182861
Apply HexK to the current goal.
L182862
Let K be given.
L182863
Assume HK.
L182863
We prove the intermediate claim HKomega: K ∈ ω.
L182865
An exact proof term for the current goal is (andEL (K ∈ ω) (eps_ K < eps) HK).
L182865
We prove the intermediate claim HepsKltEpsS: eps_ K < eps.
L182867
An exact proof term for the current goal is (andER (K ∈ ω) (eps_ K < eps) HK).
L182867
We prove the intermediate claim HepsKR: eps_ K ∈ R.
L182869
An exact proof term for the current goal is (SNoS_omega_real (eps_ K) (SNo_eps_SNoS_omega K HKomega)).
L182869
We prove the intermediate claim HepsKltEps: Rlt (eps_ K) eps.
L182871
An exact proof term for the current goal is (RltI (eps_ K) eps HepsKR HepsR HepsKltEpsS).
L182871
We prove the intermediate claim HexN: ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → ((State n) 1) 1 < eps_ K.
(*** remaining goal: find N with ((State n)1)1 < eps_ K uniformly for n>=N ***)
L182876
We prove the intermediate claim HKnat: nat_p K.
L182877
An exact proof term for the current goal is (omega_nat_p K HKomega).
L182877
We prove the intermediate claim HKcase: K = 0 ∨ ∃k : set, nat_p k ∧ K = ordsucc k.
L182879
An exact proof term for the current goal is (nat_inv K HKnat).
L182879
Apply HKcase to the current goal.
L182881
Assume HK0: K = 0.
L182881
We use 0 to witness the existential quantifier.
L182882
Apply andI to the current goal.
L182884
An exact proof term for the current goal is (nat_p_omega 0 nat_0).
L182885
Let n be given.
L182885
Assume HnO: n ∈ ω.
L182885
Assume H0sub: 0 ⊆ n.
L182886
We will prove ((State n) 1) 1 < eps_ K.
L182887
rewrite the current goal using HK0 (from left to right).
L182888
rewrite the current goal using eps_0_1 (from left to right).
L182889
An exact proof term for the current goal is (HInv_c_lt1 n HnO).
L182891
Assume Hexk: ∃k : set, nat_p k ∧ K = ordsucc k.
L182891
Apply Hexk to the current goal.
L182892
Let k be given.
L182893
Assume Hkconj: nat_p k ∧ K = ordsucc k.
L182893
We prove the intermediate claim HkNat: nat_p k.
L182895
An exact proof term for the current goal is (andEL (nat_p k) (K = ordsucc k) Hkconj).
L182895
We prove the intermediate claim HKeq: K = ordsucc k.
L182897
An exact proof term for the current goal is (andER (nat_p k) (K = ordsucc k) Hkconj).
L182897
Set N0 to be the term add_nat K K.
L182899
We use N0 to witness the existential quantifier.
(*** TODO: prove geometric decay of the scalar component and compare with eps_(ordsucc k) ***)
L182900
Apply andI to the current goal.
L182902
We prove the intermediate claim HN0O: N0 ∈ ω.
L182903
rewrite the current goal using (add_nat_add_SNo K HKomega K HKomega) (from left to right).
L182903
An exact proof term for the current goal is (add_SNo_In_omega K HKomega K HKomega).
L182904
An exact proof term for the current goal is HN0O.
L182906
Let n be given.
L182906
Assume HnO: n ∈ ω.
L182906
Assume HN0sub: N0 ⊆ n.
L182907
We prove the intermediate claim HKnat: nat_p K.
(*** show ((State n) 1) 1 < eps_ K from a dyadic bound at N0 and monotone decay ***)
L182910
An exact proof term for the current goal is (omega_nat_p K HKomega).
L182910
We prove the intermediate claim HdenS: SNo den.
L182912
An exact proof term for the current goal is (real_SNo den HdenR).
L182912
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
L182914
We prove the intermediate claim HdenLt1: den < 1.
L182916
rewrite the current goal using HdenDef (from left to right).
L182916
An exact proof term for the current goal is (RltE_lt two_thirds 1 Rlt_two_thirds_1).
L182917
Set den2 to be the term mul_SNo den den.
L182918
We prove the intermediate claim Hden2S: SNo den2.
L182920
An exact proof term for the current goal is (SNo_mul_SNo den den HdenS HdenS).
L182920
We prove the intermediate claim Hden2Lt_eps1: den2 < eps_ 1.
L182922
rewrite the current goal using HdenDef (from left to right) at position 1.
L182922
rewrite the current goal using HdenDef (from left to right) at position 2.
L182923
An exact proof term for the current goal is two_thirds_sq_lt_eps_1.
L182924
We prove the intermediate claim Hc_step: ∀m : set, nat_p m → ((State (ordsucc m)) 1) 1 = mul_SNo (((State m) 1) 1) den.
(*** scalar recursion: c_{S m} = c_m times den ***)
L182928
Let m be given.
L182928
Assume HmNat: nat_p m.
L182928
We prove the intermediate claim HS: State (ordsucc m) = StepState m (State m).
L182930
An exact proof term for the current goal is (nat_primrec_S BaseState StepState m HmNat).
L182930
rewrite the current goal using HS (from left to right).
L182931
Set st to be the term State m.
L182932
Set g to be the term st 0.
L182933
Set rpack to be the term st 1.
L182934
Set r to be the term rpack 0.
L182935
Set c to be the term rpack 1.
L182936
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L182939
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L182944
Set cNew to be the term mul_SNo c den.
L182945
We prove the intermediate claim Hdef: StepState m st = (gNew,(rNew,cNew)).
Use reflexivity.
L182947
We prove the intermediate claim HtEq: ((StepState m st) 1) 1 = cNew.
L182949
We prove the intermediate claim Hinner: (StepState m st) 1 = (rNew,cNew).
L182950
rewrite the current goal using Hdef (from left to right).
L182950
An exact proof term for the current goal is (tuple_2_1_eq gNew (rNew,cNew)).
L182951
rewrite the current goal using Hinner (from left to right).
L182952
An exact proof term for the current goal is (tuple_2_1_eq rNew cNew).
L182953
rewrite the current goal using HtEq (from left to right).
L182954
We prove the intermediate claim HcEq: c = ((State m) 1) 1.
Use reflexivity.
L182956
rewrite the current goal using HcEq (from left to right).
Use reflexivity.
L182958
We prove the intermediate claim Hc_succ_lt: ∀t : set, t ∈ ω → ((State (ordsucc t)) 1) 1 < ((State t) 1) 1.
(*** monotone decay: c_{S t + a} < c_{t + a} for t in omega ***)
L182962
Let t be given.
L182962
Assume HtO: t ∈ ω.
L182962
We prove the intermediate claim HtNat: nat_p t.
L182964
An exact proof term for the current goal is (omega_nat_p t HtO).
L182964
We prove the intermediate claim HcEq: ((State (ordsucc t)) 1) 1 = mul_SNo (((State t) 1) 1) den.
L182966
An exact proof term for the current goal is (Hc_step t HtNat).
L182966
rewrite the current goal using HcEq (from left to right).
L182967
We prove the intermediate claim HctR: ((State t) 1) 1 ∈ R.
L182969
An exact proof term for the current goal is (andEL (((State t) 1) 1 ∈ R) (0 < ((State t) 1) 1) (HInv_cpos t HtO)).
L182969
We prove the intermediate claim HctS: SNo (((State t) 1) 1).
L182971
An exact proof term for the current goal is (real_SNo (((State t) 1) 1) HctR).
L182971
We prove the intermediate claim HctPos: 0 < (((State t) 1) 1).
L182973
An exact proof term for the current goal is (andER (((State t) 1) 1 ∈ R) (0 < ((State t) 1) 1) (HInv_cpos t HtO)).
L182973
We prove the intermediate claim HmulLt: mul_SNo den (((State t) 1) 1) < ((State t) 1) 1.
L182975
An exact proof term for the current goal is (mul_SNo_Lt1_pos_Lt den (((State t) 1) 1) HdenS HctS HdenLt1 HctPos).
L182975
We prove the intermediate claim HmulEq: mul_SNo den (((State t) 1) 1) = mul_SNo (((State t) 1) 1) den.
L182977
An exact proof term for the current goal is (mul_SNo_com den (((State t) 1) 1) HdenS HctS).
L182977
rewrite the current goal using HmulEq (from right to left).
L182978
An exact proof term for the current goal is HmulLt.
L182979
We prove the intermediate claim Hc_even_lt: ∀t : set, nat_p t → ((State (add_nat t t)) 1) 1 < eps_ t.
(*** dyadic bound at even indices: c_{K+K} < eps_K ***)
L182983
Apply nat_ind to the current goal.
L182984
We will prove ((State (add_nat 0 0)) 1) 1 < eps_ 0.
L182984
rewrite the current goal using (add_nat_0R 0) (from left to right) at position 1.
L182985
rewrite the current goal using eps_0_1 (from left to right).
L182986
An exact proof term for the current goal is (HInv_c_lt1 0 (nat_p_omega 0 nat_0)).
L182988
Let t be given.
L182988
Assume HtNat: nat_p t.
L182988
Assume IH: ((State (add_nat t t)) 1) 1 < eps_ t.
L182989
We will prove ((State (add_nat (ordsucc t) (ordsucc t))) 1) 1 < eps_ (ordsucc t).
L182990
Set N to be the term add_nat t t.
L182991
We prove the intermediate claim HNnat: nat_p N.
L182993
An exact proof term for the current goal is (add_nat_p t HtNat t HtNat).
L182993
We prove the intermediate claim Hidx: add_nat (ordsucc t) (ordsucc t) = ordsucc (ordsucc N).
L182995
rewrite the current goal using (add_nat_SL t HtNat (ordsucc t) (nat_ordsucc t HtNat)) (from left to right).
L182995
rewrite the current goal using (add_nat_SR t t HtNat) (from left to right).
Use reflexivity.
L182997
rewrite the current goal using Hidx (from left to right).
L182998
We prove the intermediate claim HN0: N ∈ ω.
L183000
An exact proof term for the current goal is (nat_p_omega N HNnat).
L183000
We prove the intermediate claim HcSN: ((State (ordsucc N)) 1) 1 = mul_SNo (((State N) 1) 1) den.
L183002
An exact proof term for the current goal is (Hc_step N HNnat).
L183002
We prove the intermediate claim HcSSN: ((State (ordsucc (ordsucc N))) 1) 1 = mul_SNo (mul_SNo (((State N) 1) 1) den) den.
L183005
We prove the intermediate claim HSNnat: nat_p (ordsucc N).
L183006
An exact proof term for the current goal is (nat_ordsucc N HNnat).
L183006
rewrite the current goal using (Hc_step (ordsucc N) HSNnat) (from left to right).
L183007
rewrite the current goal using HcSN (from left to right).
Use reflexivity.
L183009
rewrite the current goal using HcSSN (from left to right).
L183010
rewrite the current goal using (mul_SNo_assoc (((State N) 1) 1) den den (real_SNo (((State N) 1) 1) (andEL (((State N) 1) 1 ∈ R) (0 < ((State N) 1) 1) (HInv_cpos N HN0))) HdenS HdenS) (from right to left).
L183013
We prove the intermediate claim Hden2Def: den2 = mul_SNo den den.
Use reflexivity.
L183015
rewrite the current goal using Hden2Def (from right to left).
L183016
We prove the intermediate claim HcNpos: 0 < ((State N) 1) 1.
L183018
An exact proof term for the current goal is (andER (((State N) 1) 1 ∈ R) (0 < ((State N) 1) 1) (HInv_cpos N HN0)).
L183018
We prove the intermediate claim HcNS: SNo (((State N) 1) 1).
L183020
An exact proof term for the current goal is (real_SNo (((State N) 1) 1) (andEL (((State N) 1) 1 ∈ R) (0 < ((State N) 1) 1) (HInv_cpos N HN0))).
L183020
rewrite the current goal using (mul_SNo_com (((State N) 1) 1) den2 HcNS Hden2S) (from left to right).
L183021
We prove the intermediate claim Heps1R: (eps_ 1) ∈ R.
L183023
An exact proof term for the current goal is (SNoS_omega_real (eps_ 1) (SNo_eps_SNoS_omega 1 (nat_p_omega 1 nat_1))).
L183023
We prove the intermediate claim Heps1S: SNo (eps_ 1).
L183025
An exact proof term for the current goal is (real_SNo (eps_ 1) Heps1R).
L183025
We prove the intermediate claim Heps1pos: 0 < eps_ 1.
L183027
An exact proof term for the current goal is (RltE_lt 0 (eps_ 1) eps_1_pos_R).
L183027
We prove the intermediate claim Hmul1: mul_SNo den2 (((State N) 1) 1) < mul_SNo (eps_ 1) (((State N) 1) 1).
L183029
An exact proof term for the current goal is (pos_mul_SNo_Lt' den2 (eps_ 1) (((State N) 1) 1) Hden2S Heps1S HcNS HcNpos Hden2Lt_eps1).
L183030
We prove the intermediate claim Heps_tR: (eps_ t) ∈ R.
L183032
An exact proof term for the current goal is (SNoS_omega_real (eps_ t) (SNo_eps_SNoS_omega t (nat_p_omega t HtNat))).
L183032
We prove the intermediate claim Heps_tS: SNo (eps_ t).
L183034
An exact proof term for the current goal is (real_SNo (eps_ t) Heps_tR).
L183034
We prove the intermediate claim Hmul2: mul_SNo (eps_ 1) (((State N) 1) 1) < mul_SNo (eps_ 1) (eps_ t).
L183036
We prove the intermediate claim HcomL: mul_SNo (eps_ 1) (((State N) 1) 1) = mul_SNo (((State N) 1) 1) (eps_ 1).
L183037
An exact proof term for the current goal is (mul_SNo_com (eps_ 1) (((State N) 1) 1) Heps1S HcNS).
L183037
We prove the intermediate claim HcomR: mul_SNo (eps_ 1) (eps_ t) = mul_SNo (eps_ t) (eps_ 1).
L183039
An exact proof term for the current goal is (mul_SNo_com (eps_ 1) (eps_ t) Heps1S Heps_tS).
L183039
rewrite the current goal using HcomL (from left to right).
L183040
rewrite the current goal using HcomR (from left to right).
L183041
An exact proof term for the current goal is (pos_mul_SNo_Lt' (((State N) 1) 1) (eps_ t) (eps_ 1) HcNS Heps_tS Heps1S Heps1pos IH).
L183043
We prove the intermediate claim HmulTra: mul_SNo den2 (((State N) 1) 1) < mul_SNo (eps_ 1) (eps_ t).
L183045
An exact proof term for the current goal is (SNoLt_tra (mul_SNo den2 (((State N) 1) 1)) (mul_SNo (eps_ 1) (((State N) 1) 1)) (mul_SNo (eps_ 1) (eps_ t)) (SNo_mul_SNo den2 (((State N) 1) 1) Hden2S HcNS) (SNo_mul_SNo (eps_ 1) (((State N) 1) 1) Heps1S HcNS) (SNo_mul_SNo (eps_ 1) (eps_ t) Heps1S Heps_tS) Hmul1 Hmul2).
L183051
We prove the intermediate claim HepsEq: mul_SNo (eps_ 1) (eps_ t) = eps_ (ordsucc t).
L183053
rewrite the current goal using (mul_SNo_eps_eps_add_SNo 1 (nat_p_omega 1 nat_1) t (nat_p_omega t HtNat)) (from left to right).
L183053
We prove the intermediate claim Hordt: ordinal t.
L183055
An exact proof term for the current goal is (nat_p_ordinal t HtNat).
L183055
rewrite the current goal using (ordinal_ordsucc_SNo_eq t Hordt) (from right to left).
Use reflexivity.
L183057
rewrite the current goal using HepsEq (from right to left).
L183058
An exact proof term for the current goal is HmulTra.
L183059
We prove the intermediate claim HcN0: ((State N0) 1) 1 < eps_ K.
L183061
An exact proof term for the current goal is (Hc_even_lt K HKnat).
L183061
We prove the intermediate claim HnNat: nat_p n.
(*** show n is N0 plus a natural shift and reduce to k=0 or successor ***)
L183064
An exact proof term for the current goal is (omega_nat_p n HnO).
L183064
We prove the intermediate claim HN0O: N0 ∈ ω.
L183066
rewrite the current goal using (add_nat_add_SNo K HKomega K HKomega) (from left to right).
L183066
An exact proof term for the current goal is (add_SNo_In_omega K HKomega K HKomega).
L183067
We prove the intermediate claim HN0Nat: nat_p N0.
L183069
An exact proof term for the current goal is (omega_nat_p N0 HN0O).
L183069
We prove the intermediate claim Hexk: ∃k : set, nat_p k ∧ n = add_nat k N0.
L183071
An exact proof term for the current goal is (nat_Subq_add_ex N0 HN0Nat n HnNat HN0sub).
L183071
Apply Hexk to the current goal.
L183072
Let k1 be given.
L183073
Assume Hk1.
L183073
We prove the intermediate claim Hk1Nat: nat_p k1.
L183075
An exact proof term for the current goal is (andEL (nat_p k1) (n = add_nat k1 N0) Hk1).
L183075
We prove the intermediate claim HnEq: n = add_nat k1 N0.
L183077
An exact proof term for the current goal is (andER (nat_p k1) (n = add_nat k1 N0) Hk1).
L183077
rewrite the current goal using HnEq (from left to right).
L183078
We prove the intermediate claim Hkcase: k1 = 0 ∨ ∃j : set, nat_p j ∧ k1 = ordsucc j.
L183080
An exact proof term for the current goal is (nat_inv k1 Hk1Nat).
L183080
Apply Hkcase to the current goal.
L183082
Assume Hk10: k1 = 0.
L183082
rewrite the current goal using Hk10 (from left to right).
L183083
rewrite the current goal using (add_nat_0L N0 HN0Nat) (from left to right).
L183084
An exact proof term for the current goal is HcN0.
L183086
Assume Hk1S: ∃j : set, nat_p j ∧ k1 = ordsucc j.
L183086
Apply Hk1S to the current goal.
L183087
Let j be given.
L183088
Assume Hj.
L183088
We prove the intermediate claim HjNat: nat_p j.
L183090
An exact proof term for the current goal is (andEL (nat_p j) (k1 = ordsucc j) Hj).
L183090
We prove the intermediate claim Hk1Eq: k1 = ordsucc j.
L183092
An exact proof term for the current goal is (andER (nat_p j) (k1 = ordsucc j) Hj).
L183092
rewrite the current goal using Hk1Eq (from left to right).
L183093
We prove the intermediate claim Hdec: ∀j : set, nat_p j → ((State (add_nat (ordsucc j) N0)) 1) 1 < ((State N0) 1) 1.
L183096
Apply nat_ind to the current goal.
L183097
We will prove ((State (add_nat (ordsucc 0) N0)) 1) 1 < ((State N0) 1) 1.
L183097
rewrite the current goal using (add_nat_SL 0 nat_0 N0 HN0Nat) (from left to right).
L183098
rewrite the current goal using (add_nat_0L N0 HN0Nat) (from left to right).
L183099
An exact proof term for the current goal is (Hc_succ_lt N0 HN0O).
L183101
Let j be given.
L183101
Assume HjNat: nat_p j.
L183101
Assume IHj: ((State (add_nat (ordsucc j) N0)) 1) 1 < ((State N0) 1) 1.
L183102
We will prove ((State (add_nat (ordsucc (ordsucc j)) N0)) 1) 1 < ((State N0) 1) 1.
L183103
rewrite the current goal using (add_nat_SL (ordsucc j) (nat_ordsucc j HjNat) N0 HN0Nat) (from left to right).
L183104
We prove the intermediate claim HstepLt: ((State (ordsucc (add_nat (ordsucc j) N0))) 1) 1 < ((State (add_nat (ordsucc j) N0)) 1) 1.
L183106
An exact proof term for the current goal is (Hc_succ_lt (add_nat (ordsucc j) N0) (nat_p_omega (add_nat (ordsucc j) N0) (add_nat_p (ordsucc j) (nat_ordsucc j HjNat) N0 HN0Nat))).
L183108
We prove the intermediate claim HaO: ordsucc (add_nat (ordsucc j) N0) ∈ ω.
L183110
An exact proof term for the current goal is (nat_p_omega (ordsucc (add_nat (ordsucc j) N0)) (nat_ordsucc (add_nat (ordsucc j) N0) (add_nat_p (ordsucc j) (nat_ordsucc j HjNat) N0 HN0Nat))).
L183112
We prove the intermediate claim HbO: add_nat (ordsucc j) N0 ∈ ω.
L183114
An exact proof term for the current goal is (nat_p_omega (add_nat (ordsucc j) N0) (add_nat_p (ordsucc j) (nat_ordsucc j HjNat) N0 HN0Nat)).
L183115
We prove the intermediate claim HaR: ((State (ordsucc (add_nat (ordsucc j) N0))) 1) 1 ∈ R.
L183117
An exact proof term for the current goal is (andEL (((State (ordsucc (add_nat (ordsucc j) N0))) 1) 1 ∈ R) (0 < ((State (ordsucc (add_nat (ordsucc j) N0))) 1) 1) (HInv_cpos (ordsucc (add_nat (ordsucc j) N0)) HaO)).
L183119
We prove the intermediate claim HbR: ((State (add_nat (ordsucc j) N0)) 1) 1 ∈ R.
L183121
An exact proof term for the current goal is (andEL (((State (add_nat (ordsucc j) N0)) 1) 1 ∈ R) (0 < ((State (add_nat (ordsucc j) N0)) 1) 1) (HInv_cpos (add_nat (ordsucc j) N0) HbO)).
L183123
We prove the intermediate claim HcR0: ((State N0) 1) 1 ∈ R.
L183125
An exact proof term for the current goal is (andEL (((State N0) 1) 1 ∈ R) (0 < ((State N0) 1) 1) (HInv_cpos N0 HN0O)).
L183125
We prove the intermediate claim HaS: SNo (((State (ordsucc (add_nat (ordsucc j) N0))) 1) 1).
L183127
An exact proof term for the current goal is (real_SNo (((State (ordsucc (add_nat (ordsucc j) N0))) 1) 1) HaR).
L183127
We prove the intermediate claim HbS: SNo (((State (add_nat (ordsucc j) N0)) 1) 1).
L183129
An exact proof term for the current goal is (real_SNo (((State (add_nat (ordsucc j) N0)) 1) 1) HbR).
L183129
We prove the intermediate claim HcS0: SNo (((State N0) 1) 1).
L183131
An exact proof term for the current goal is (real_SNo (((State N0) 1) 1) HcR0).
L183131
An exact proof term for the current goal is (SNoLt_tra (((State (ordsucc (add_nat (ordsucc j) N0))) 1) 1) (((State (add_nat (ordsucc j) N0)) 1) 1) (((State N0) 1) 1) HaS HbS HcS0 HstepLt IHj).
L183135
We prove the intermediate claim HltN0: ((State (add_nat (ordsucc j) N0)) 1) 1 < ((State N0) 1) 1.
L183137
An exact proof term for the current goal is (Hdec j HjNat).
L183137
We prove the intermediate claim HxO: add_nat (ordsucc j) N0 ∈ ω.
L183139
An exact proof term for the current goal is (nat_p_omega (add_nat (ordsucc j) N0) (add_nat_p (ordsucc j) (nat_ordsucc j HjNat) N0 HN0Nat)).
L183140
We prove the intermediate claim HxR: ((State (add_nat (ordsucc j) N0)) 1) 1 ∈ R.
L183142
An exact proof term for the current goal is (andEL (((State (add_nat (ordsucc j) N0)) 1) 1 ∈ R) (0 < ((State (add_nat (ordsucc j) N0)) 1) 1) (HInv_cpos (add_nat (ordsucc j) N0) HxO)).
L183144
We prove the intermediate claim HxS: SNo (((State (add_nat (ordsucc j) N0)) 1) 1).
L183146
An exact proof term for the current goal is (real_SNo (((State (add_nat (ordsucc j) N0)) 1) 1) HxR).
L183146
We prove the intermediate claim HN0R: ((State N0) 1) 1 ∈ R.
L183148
An exact proof term for the current goal is (andEL (((State N0) 1) 1 ∈ R) (0 < ((State N0) 1) 1) (HInv_cpos N0 HN0O)).
L183148
We prove the intermediate claim HN0S: SNo (((State N0) 1) 1).
L183150
An exact proof term for the current goal is (real_SNo (((State N0) 1) 1) HN0R).
L183150
We prove the intermediate claim HepsKS: SNo (eps_ K).
L183152
An exact proof term for the current goal is (real_SNo (eps_ K) HepsKR).
L183152
An exact proof term for the current goal is (SNoLt_tra (((State (add_nat (ordsucc j) N0)) 1) 1) (((State N0) 1) 1) (eps_ K) HxS HN0S HepsKS HltN0 HcN0).
L183156
Apply HexN to the current goal.
L183157
Let N be given.
L183158
Assume HN.
L183158
We use N to witness the existential quantifier.
L183159
Apply andI to the current goal.
L183161
An exact proof term for the current goal is (andEL (N ∈ ω) (∀n : set, n ∈ ω → N ⊆ n → ((State n) 1) 1 < eps_ K) HN).
L183164
Let n be given.
L183164
Assume HnO: n ∈ ω.
L183164
Assume HNsub: N ⊆ n.
L183165
We prove the intermediate claim HNprop: ∀t : set, t ∈ ω → N ⊆ t → ((State t) 1) 1 < eps_ K.
L183168
An exact proof term for the current goal is (andER (N ∈ ω) (∀n : set, n ∈ ω → N ⊆ n → ((State n) 1) 1 < eps_ K) HN).
L183170
We prove the intermediate claim HcLtEpsK: ((State n) 1) 1 < eps_ K.
L183172
An exact proof term for the current goal is (HNprop n HnO HNsub).
L183172
We prove the intermediate claim HcR: ((State n) 1) 1 ∈ R.
L183174
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L183174
We prove the intermediate claim HcS: SNo (((State n) 1) 1).
L183176
An exact proof term for the current goal is (real_SNo (((State n) 1) 1) HcR).
L183176
We prove the intermediate claim HepsKS: SNo (eps_ K).
L183178
An exact proof term for the current goal is (real_SNo (eps_ K) HepsKR).
L183178
We prove the intermediate claim HepsS0: SNo eps.
L183180
An exact proof term for the current goal is (real_SNo eps HepsR).
L183180
An exact proof term for the current goal is (SNoLt_tra (((State n) 1) 1) (eps_ K) eps HcS HepsKS HepsS0 HcLtEpsK HepsKltEpsS).
L183182
Apply Hex_c_small to the current goal.
L183183
Let N be given.
L183184
Assume HNpair.
L183184
We use N to witness the existential quantifier.
L183185
Apply andI to the current goal.
L183187
An exact proof term for the current goal is (andEL (N ∈ ω) (∀n : set, n ∈ ω → N ⊆ n → ((State n) 1) 1 < eps) HNpair).
L183190
Let n be given.
L183190
Assume HnO: n ∈ ω.
L183190
Assume HNsub: N ⊆ n.
L183191
We will prove Rlt (apply_fun R_bounded_metric (apply_fun seq2 n,lim)) eps.
L183192
We prove the intermediate claim Hseq2nR: apply_fun seq2 n ∈ R.
L183194
An exact proof term for the current goal is (Hseq2On n HnO).
L183194
We prove the intermediate claim HlimR: lim ∈ R.
L183196
An exact proof term for the current goal is (Hf_R x HxA).
L183196
We prove the intermediate claim Hpair: (apply_fun seq2 n,lim) ∈ setprod R R.
L183198
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun seq2 n) lim Hseq2nR HlimR).
L183198
We prove the intermediate claim HdefM: R_bounded_metric = graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)).
Use reflexivity.
L183201
rewrite the current goal using HdefM (from left to right).
L183202
rewrite the current goal using (apply_fun_graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)) (apply_fun seq2 n,lim) Hpair) (from left to right).
L183204
rewrite the current goal using (tuple_2_0_eq (apply_fun seq2 n) lim) (from left to right).
L183205
rewrite the current goal using (tuple_2_1_eq (apply_fun seq2 n) lim) (from left to right).
L183206
Set a to be the term apply_fun seq2 n.
L183207
Set b to be the term lim.
L183208
Set t to be the term add_SNo a (minus_SNo b).
L183209
We prove the intermediate claim HmbR: minus_SNo b ∈ R.
L183211
An exact proof term for the current goal is (real_minus_SNo b HlimR).
L183211
We prove the intermediate claim HtR: t ∈ R.
L183213
An exact proof term for the current goal is (real_add_SNo a Hseq2nR (minus_SNo b) HmbR).
L183213
We prove the intermediate claim HtS: SNo t.
L183215
An exact proof term for the current goal is (real_SNo t HtR).
L183215
We prove the intermediate claim HaS: SNo a.
L183217
An exact proof term for the current goal is (real_SNo a Hseq2nR).
L183217
We prove the intermediate claim HbS: SNo b.
L183219
An exact proof term for the current goal is (real_SNo b HlimR).
L183219
We prove the intermediate claim HabsEq: abs_SNo t = abs_SNo (add_SNo b (minus_SNo a)).
L183221
An exact proof term for the current goal is (abs_SNo_dist_swap a b HaS HbS).
L183221
We prove the intermediate claim HaEq: a = apply_fun ((State n) 0) x.
L183223
We prove the intermediate claim Hseq2def: seq2 = graph ω (λn0 : set ⇒ apply_fun ((State n0) 0) x).
Use reflexivity.
L183224
rewrite the current goal using Hseq2def (from left to right).
L183225
rewrite the current goal using (apply_fun_graph ω (λn0 : set ⇒ apply_fun ((State n0) 0) x) n HnO) (from left to right).
Use reflexivity.
L183227
We prove the intermediate claim Hident: add_SNo a (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) = b.
L183231
rewrite the current goal using HaEq (from left to right).
L183231
An exact proof term for the current goal is (HInv_residual_identity_A n HnO x HxA).
L183232
We prove the intermediate claim HrcEq: add_SNo b (minus_SNo a) = mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1).
L183236
rewrite the current goal using Hident (from right to left) at position 1.
L183236
Set rc to be the term mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1).
L183237
We prove the intermediate claim HrcR: rc ∈ R.
L183239
We prove the intermediate claim Hrfun: continuous_map A Ta I Ti (((State n) 1) 0).
L183240
An exact proof term for the current goal is (HInv_r_contI n HnO).
L183240
We prove the intermediate claim Hrfun_on: function_on (((State n) 1) 0) A I.
L183242
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti (((State n) 1) 0) Hrfun).
L183242
We prove the intermediate claim HrxI: apply_fun (((State n) 1) 0) x ∈ I.
L183244
An exact proof term for the current goal is (Hrfun_on x HxA).
L183244
We prove the intermediate claim HrxR: apply_fun (((State n) 1) 0) x ∈ R.
L183246
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun (((State n) 1) 0) x) HrxI).
L183246
We prove the intermediate claim HcR: ((State n) 1) 1 ∈ R.
L183248
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L183248
An exact proof term for the current goal is (real_mul_SNo (apply_fun (((State n) 1) 0) x) HrxR (((State n) 1) 1) HcR).
L183249
We prove the intermediate claim HrcS: SNo rc.
L183251
An exact proof term for the current goal is (real_SNo rc HrcR).
L183251
We prove the intermediate claim HaS0: SNo a.
L183253
An exact proof term for the current goal is (real_SNo a Hseq2nR).
L183253
rewrite the current goal using (add_SNo_com a rc HaS0 HrcS) (from left to right) at position 1.
L183254
An exact proof term for the current goal is (add_SNo_minus_R2 rc a HrcS HaS0).
L183255
We prove the intermediate claim HcR: ((State n) 1) 1 ∈ R.
L183257
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L183257
We prove the intermediate claim HcS: SNo (((State n) 1) 1).
L183259
An exact proof term for the current goal is (real_SNo (((State n) 1) 1) HcR).
L183259
We prove the intermediate claim HcLtEps: ((State n) 1) 1 < eps.
L183261
An exact proof term for the current goal is (andER (N ∈ ω) (∀n0 : set, n0 ∈ ω → N ⊆ n0 → ((State n0) 1) 1 < eps) HNpair n HnO HNsub).
L183263
We prove the intermediate claim Hcpos: 0 < ((State n) 1) 1.
L183265
An exact proof term for the current goal is (andER (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L183265
We prove the intermediate claim H0le_c: 0 ≤ ((State n) 1) 1.
L183267
An exact proof term for the current goal is (SNoLtLe 0 (((State n) 1) 1) Hcpos).
L183267
We prove the intermediate claim Hrfun: continuous_map A Ta I Ti (((State n) 1) 0).
L183269
An exact proof term for the current goal is (HInv_r_contI n HnO).
L183269
We prove the intermediate claim Hrfun_on: function_on (((State n) 1) 0) A I.
L183271
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti (((State n) 1) 0) Hrfun).
L183271
We prove the intermediate claim HrxI: apply_fun (((State n) 1) 0) x ∈ I.
L183273
An exact proof term for the current goal is (Hrfun_on x HxA).
L183273
We prove the intermediate claim HrxR: apply_fun (((State n) 1) 0) x ∈ R.
L183275
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun (((State n) 1) 0) x) HrxI).
L183275
We prove the intermediate claim HprodI: mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1) ∈ closed_interval (minus_SNo (((State n) 1) 1)) (((State n) 1) 1).
L183278
An exact proof term for the current goal is (mul_nonneg_closed_interval_minus1_1 (apply_fun (((State n) 1) 0) x) (((State n) 1) 1) HrxI HcR H0le_c).
L183279
We prove the intermediate claim Hmcr: (minus_SNo (((State n) 1) 1)) ∈ R.
L183281
An exact proof term for the current goal is (real_minus_SNo (((State n) 1) 1) HcR).
L183281
We prove the intermediate claim HprodR: mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1) ∈ R.
L183283
An exact proof term for the current goal is (real_mul_SNo (apply_fun (((State n) 1) 0) x) HrxR (((State n) 1) 1) HcR).
L183283
We prove the intermediate claim HprodS: SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)).
L183285
An exact proof term for the current goal is (real_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) HprodR).
L183285
We prove the intermediate claim Hbounds: Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) ∧ Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1).
L183289
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo (((State n) 1) 1)) (((State n) 1) 1) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) Hmcr HcR HprodI).
L183290
We prove the intermediate claim HloRle: Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)).
L183293
An exact proof term for the current goal is (andEL (Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) (Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1)) Hbounds).
L183296
We prove the intermediate claim HhiRle: Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1).
L183299
An exact proof term for the current goal is (andER (Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) (Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1)) Hbounds).
L183302
We prove the intermediate claim HloLe: minus_SNo (((State n) 1) 1) ≤ mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1).
L183304
An exact proof term for the current goal is (SNoLe_of_Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) HloRle).
L183306
We prove the intermediate claim HhiLe: mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1) ≤ ((State n) 1) 1.
L183308
An exact proof term for the current goal is (SNoLe_of_Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1) HhiRle).
L183311
We prove the intermediate claim HabsLe: abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) ≤ ((State n) 1) 1.
L183313
An exact proof term for the current goal is (abs_SNo_Le_of_bounds (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1) HprodS HcS HloLe HhiLe).
L183319
We prove the intermediate claim HabsR: abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) ∈ R.
L183321
An exact proof term for the current goal is (abs_SNo_in_R (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) HprodR).
L183321
We prove the intermediate claim HabsS: SNo (abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))).
L183323
An exact proof term for the current goal is (real_SNo (abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) HabsR).
L183323
We prove the intermediate claim HabsLtEps: abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) < eps.
L183326
An exact proof term for the current goal is (SNoLeLt_tra (abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) (((State n) 1) 1) eps HabsS HcS HepsS HabsLe HcLtEps).
L183334
We prove the intermediate claim HabstLtEps: abs_SNo t < eps.
L183336
rewrite the current goal using HabsEq (from left to right).
L183336
rewrite the current goal using HrcEq (from left to right).
L183337
An exact proof term for the current goal is HabsLtEps.
L183338
An exact proof term for the current goal is (abs_lt_lt1_imp_R_bounded_distance_lt a b eps Hseq2nR HlimR HepsR HepsLt1 HabstLtEps).
L183341
Assume HepsEq1: eps = 1.
L183341
rewrite the current goal using HepsEq1 (from left to right).
L183343
We use 0 to witness the existential quantifier.
(*** TODO: show eventually R_bounded_distance < 1 using abs difference < 1 from the residual bound on A. ***)
L183344
Apply andI to the current goal.
L183346
An exact proof term for the current goal is (nat_p_omega 0 nat_0).
L183347
Let n be given.
L183347
Assume HnO: n ∈ ω.
L183347
Assume H0sub: 0 ⊆ n.
L183348
We will prove Rlt (apply_fun R_bounded_metric (apply_fun seq2 n,lim)) 1.
L183349
We prove the intermediate claim Hseq2nR: apply_fun seq2 n ∈ R.
L183351
An exact proof term for the current goal is (Hseq2On n HnO).
L183351
We prove the intermediate claim HlimR: lim ∈ R.
L183353
An exact proof term for the current goal is (Hf_R x HxA).
L183353
We prove the intermediate claim Hpair: (apply_fun seq2 n,lim) ∈ setprod R R.
L183355
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun seq2 n) lim Hseq2nR HlimR).
L183355
We prove the intermediate claim HdefM: R_bounded_metric = graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)).
Use reflexivity.
L183358
rewrite the current goal using HdefM (from left to right).
L183359
rewrite the current goal using (apply_fun_graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)) (apply_fun seq2 n,lim) Hpair) (from left to right).
L183361
rewrite the current goal using (tuple_2_0_eq (apply_fun seq2 n) lim) (from left to right).
L183362
rewrite the current goal using (tuple_2_1_eq (apply_fun seq2 n) lim) (from left to right).
L183363
Set a to be the term apply_fun seq2 n.
L183364
Set b to be the term lim.
L183365
Set t to be the term add_SNo a (minus_SNo b).
L183366
We prove the intermediate claim HmbR: minus_SNo b ∈ R.
L183368
An exact proof term for the current goal is (real_minus_SNo b HlimR).
L183368
We prove the intermediate claim HtR: t ∈ R.
L183370
An exact proof term for the current goal is (real_add_SNo a Hseq2nR (minus_SNo b) HmbR).
L183370
We prove the intermediate claim HtS: SNo t.
L183372
An exact proof term for the current goal is (real_SNo t HtR).
L183372
We prove the intermediate claim HaS: SNo a.
L183374
An exact proof term for the current goal is (real_SNo a Hseq2nR).
L183374
We prove the intermediate claim HbS: SNo b.
L183376
An exact proof term for the current goal is (real_SNo b HlimR).
L183376
We prove the intermediate claim HabsEq: abs_SNo t = abs_SNo (add_SNo b (minus_SNo a)).
L183378
An exact proof term for the current goal is (abs_SNo_dist_swap a b HaS HbS).
L183378
We prove the intermediate claim HaEq: a = apply_fun ((State n) 0) x.
L183380
We prove the intermediate claim Hseq2def: seq2 = graph ω (λn0 : set ⇒ apply_fun ((State n0) 0) x).
Use reflexivity.
L183381
rewrite the current goal using Hseq2def (from left to right).
L183382
rewrite the current goal using (apply_fun_graph ω (λn0 : set ⇒ apply_fun ((State n0) 0) x) n HnO) (from left to right).
Use reflexivity.
L183384
We prove the intermediate claim Hident: add_SNo a (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) = b.
L183388
rewrite the current goal using HaEq (from left to right).
L183388
An exact proof term for the current goal is (HInv_residual_identity_A n HnO x HxA).
L183389
We prove the intermediate claim HrcEq: add_SNo b (minus_SNo a) = mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1).
L183393
rewrite the current goal using Hident (from right to left) at position 1.
L183393
Set rc to be the term mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1).
L183394
We prove the intermediate claim HrcR: rc ∈ R.
L183396
We prove the intermediate claim Hrfun: continuous_map A Ta I Ti (((State n) 1) 0).
L183397
An exact proof term for the current goal is (HInv_r_contI n HnO).
L183397
We prove the intermediate claim Hrfun_on: function_on (((State n) 1) 0) A I.
L183399
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti (((State n) 1) 0) Hrfun).
L183399
We prove the intermediate claim HrxI: apply_fun (((State n) 1) 0) x ∈ I.
L183401
An exact proof term for the current goal is (Hrfun_on x HxA).
L183401
We prove the intermediate claim HrxR: apply_fun (((State n) 1) 0) x ∈ R.
L183403
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun (((State n) 1) 0) x) HrxI).
L183403
We prove the intermediate claim HcR: ((State n) 1) 1 ∈ R.
L183405
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L183405
An exact proof term for the current goal is (real_mul_SNo (apply_fun (((State n) 1) 0) x) HrxR (((State n) 1) 1) HcR).
L183406
We prove the intermediate claim HrcS: SNo rc.
L183408
An exact proof term for the current goal is (real_SNo rc HrcR).
L183408
We prove the intermediate claim HaS0: SNo a.
L183410
An exact proof term for the current goal is (real_SNo a Hseq2nR).
L183410
rewrite the current goal using (add_SNo_com a rc HaS0 HrcS) (from left to right) at position 1.
L183411
An exact proof term for the current goal is (add_SNo_minus_R2 rc a HrcS HaS0).
L183412
We prove the intermediate claim HcR: ((State n) 1) 1 ∈ R.
L183414
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L183414
We prove the intermediate claim HcS: SNo (((State n) 1) 1).
L183416
An exact proof term for the current goal is (real_SNo (((State n) 1) 1) HcR).
L183416
We prove the intermediate claim HcLt1: ((State n) 1) 1 < 1.
L183418
An exact proof term for the current goal is (HInv_c_lt1 n HnO).
L183418
We prove the intermediate claim Hcpos: 0 < ((State n) 1) 1.
L183420
An exact proof term for the current goal is (andER (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L183420
We prove the intermediate claim H0le_c: 0 ≤ ((State n) 1) 1.
L183422
An exact proof term for the current goal is (SNoLtLe 0 (((State n) 1) 1) Hcpos).
L183422
We prove the intermediate claim Hrfun: continuous_map A Ta I Ti (((State n) 1) 0).
L183424
An exact proof term for the current goal is (HInv_r_contI n HnO).
L183424
We prove the intermediate claim Hrfun_on: function_on (((State n) 1) 0) A I.
L183426
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti (((State n) 1) 0) Hrfun).
L183426
We prove the intermediate claim HrxI: apply_fun (((State n) 1) 0) x ∈ I.
L183428
An exact proof term for the current goal is (Hrfun_on x HxA).
L183428
We prove the intermediate claim HrxR: apply_fun (((State n) 1) 0) x ∈ R.
L183430
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun (((State n) 1) 0) x) HrxI).
L183430
We prove the intermediate claim HrxS: SNo (apply_fun (((State n) 1) 0) x).
L183432
An exact proof term for the current goal is (real_SNo (apply_fun (((State n) 1) 0) x) HrxR).
L183432
We prove the intermediate claim HprodI: mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1) ∈ closed_interval (minus_SNo (((State n) 1) 1)) (((State n) 1) 1).
L183435
An exact proof term for the current goal is (mul_nonneg_closed_interval_minus1_1 (apply_fun (((State n) 1) 0) x) (((State n) 1) 1) HrxI HcR H0le_c).
L183436
We prove the intermediate claim Hmcr: (minus_SNo (((State n) 1) 1)) ∈ R.
L183438
An exact proof term for the current goal is (real_minus_SNo (((State n) 1) 1) HcR).
L183438
We prove the intermediate claim HprodR: mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1) ∈ R.
L183440
An exact proof term for the current goal is (real_mul_SNo (apply_fun (((State n) 1) 0) x) HrxR (((State n) 1) 1) HcR).
L183440
We prove the intermediate claim HprodS: SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)).
L183442
An exact proof term for the current goal is (real_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) HprodR).
L183442
We prove the intermediate claim Hbounds: Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) ∧ Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1).
L183446
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo (((State n) 1) 1)) (((State n) 1) 1) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) Hmcr HcR HprodI).
L183447
We prove the intermediate claim HloRle: Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)).
L183450
An exact proof term for the current goal is (andEL (Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) (Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1)) Hbounds).
L183453
We prove the intermediate claim HhiRle: Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1).
L183456
An exact proof term for the current goal is (andER (Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) (Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1)) Hbounds).
L183459
We prove the intermediate claim HloLe: minus_SNo (((State n) 1) 1) ≤ mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1).
L183461
An exact proof term for the current goal is (SNoLe_of_Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) HloRle).
L183463
We prove the intermediate claim HhiLe: mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1) ≤ ((State n) 1) 1.
L183465
An exact proof term for the current goal is (SNoLe_of_Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1) HhiRle).
L183468
We prove the intermediate claim HabsLe: abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) ≤ ((State n) 1) 1.
L183470
An exact proof term for the current goal is (abs_SNo_Le_of_bounds (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1) HprodS HcS HloLe HhiLe).
L183476
We prove the intermediate claim HabsR: abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) ∈ R.
L183478
An exact proof term for the current goal is (abs_SNo_in_R (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) HprodR).
L183478
We prove the intermediate claim HabsS: SNo (abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))).
L183480
An exact proof term for the current goal is (real_SNo (abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) HabsR).
L183480
We prove the intermediate claim HabsLt1: abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) < 1.
L183483
An exact proof term for the current goal is (SNoLeLt_tra (abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) (((State n) 1) 1) 1 HabsS HcS SNo_1 HabsLe HcLt1).
L183491
We prove the intermediate claim HabsLt1R: Rlt (abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) 1.
L183493
An exact proof term for the current goal is (RltI (abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) 1 HabsR real_1 HabsLt1).
L183498
We prove the intermediate claim HabstLt1: Rlt (abs_SNo t) 1.
L183500
rewrite the current goal using HabsEq (from left to right).
L183500
rewrite the current goal using HrcEq (from left to right).
L183501
An exact proof term for the current goal is HabsLt1R.
L183502
We prove the intermediate claim Hbddef: R_bounded_distance a b = If_i (Rlt (abs_SNo t) 1) (abs_SNo t) 1.
Use reflexivity.
L183505
rewrite the current goal using Hbddef (from left to right).
L183506
rewrite the current goal using (If_i_1 (Rlt (abs_SNo t) 1) (abs_SNo t) 1 HabstLt1) (from left to right).
L183507
An exact proof term for the current goal is HabstLt1.
L183509
Assume H1LtEpsS: 1 < eps.
L183509
We prove the intermediate claim H1LtEps: Rlt 1 eps.
L183511
An exact proof term for the current goal is (RltI 1 eps real_1 HepsR H1LtEpsS).
L183511
We use 0 to witness the existential quantifier.
L183512
Apply andI to the current goal.
L183514
An exact proof term for the current goal is (nat_p_omega 0 nat_0).
L183515
Let n be given.
L183515
Assume HnO: n ∈ ω.
L183515
Assume H0sub: 0 ⊆ n.
L183516
We will prove Rlt (apply_fun R_bounded_metric (apply_fun seq2 n,lim)) eps.
L183517
We prove the intermediate claim Hseq2nR: apply_fun seq2 n ∈ R.
L183519
An exact proof term for the current goal is (Hseq2On n HnO).
L183519
We prove the intermediate claim HlimR: lim ∈ R.
L183521
An exact proof term for the current goal is (Hf_R x HxA).
L183521
We prove the intermediate claim Hpair: (apply_fun seq2 n,lim) ∈ setprod R R.
L183523
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun seq2 n) lim Hseq2nR HlimR).
L183523
We prove the intermediate claim HdefM: R_bounded_metric = graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)).
Use reflexivity.
L183526
rewrite the current goal using HdefM (from left to right).
L183527
rewrite the current goal using (apply_fun_graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)) (apply_fun seq2 n,lim) Hpair) (from left to right).
L183529
rewrite the current goal using (tuple_2_0_eq (apply_fun seq2 n) lim) (from left to right).
L183530
rewrite the current goal using (tuple_2_1_eq (apply_fun seq2 n) lim) (from left to right).
L183531
We prove the intermediate claim Hle1: Rle (R_bounded_distance (apply_fun seq2 n) lim) 1.
L183533
An exact proof term for the current goal is (R_bounded_distance_le_1 (apply_fun seq2 n) lim Hseq2nR HlimR).
L183533
An exact proof term for the current goal is (Rle_Rlt_tra (R_bounded_distance (apply_fun seq2 n) lim) 1 eps Hle1 H1LtEps).
L183534
An exact proof term for the current goal is (sequence_converges_metric_imp_converges_to_metric_topology R R_bounded_metric seq2 lim HconvM).
L183536
An exact proof term for the current goal is Hseq2_conv.
L183538
Let eps be given.
L183539
Assume HepsR: eps ∈ R.
L183539
Assume HepsPos: Rlt 0 eps.
L183539
We will prove ∃N : set, N ∈ ω ∧ ∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) eps.
(*** uniform Cauchy (to be proved from a geometric tail estimate for the correction terms) ***)
L183543
We prove the intermediate claim HepsS: SNo eps.
(*** TODO: derive from the recursive definition of State and the bounds u_of(r) :e I0 with coefficients (2/3)^k. ***)
(*** TODO: pick N using an eps_N tail bound smaller than eps and compare in the bounded metric. ***)
L183547
An exact proof term for the current goal is (real_SNo eps HepsR).
L183547
We prove the intermediate claim Huc_small: ∀eps0 : set, eps0 ∈ R → Rlt 0 eps0 → Rlt eps0 1 → ∃N : set, N ∈ ω ∧ ∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) eps0.
L183554
Let eps0 be given.
L183554
Assume Heps0R: eps0 ∈ R.
L183554
Assume Heps0Pos: Rlt 0 eps0.
L183554
Assume Heps0Lt1: Rlt eps0 1.
L183554
We prove the intermediate claim Heps0S: SNo eps0.
L183556
An exact proof term for the current goal is (real_SNo eps0 Heps0R).
L183556
Set eta to be the term mul_SNo eps0 (eps_ 1).
L183557
We prove the intermediate claim HetaR: eta ∈ R.
L183559
An exact proof term for the current goal is (real_mul_SNo eps0 Heps0R (eps_ 1) eps_1_in_R).
L183559
We prove the intermediate claim HetaS: SNo eta.
L183561
An exact proof term for the current goal is (real_SNo eta HetaR).
L183561
We prove the intermediate claim Heps0PosS: 0 < eps0.
L183563
An exact proof term for the current goal is (RltE_lt 0 eps0 Heps0Pos).
L183563
We prove the intermediate claim Heps1S: SNo (eps_ 1).
L183565
An exact proof term for the current goal is (real_SNo (eps_ 1) eps_1_in_R).
L183565
We prove the intermediate claim Heps1PosS: 0 < eps_ 1.
L183567
An exact proof term for the current goal is (RltE_lt 0 (eps_ 1) eps_1_pos_R).
L183567
We prove the intermediate claim HetaPosS: 0 < eta.
L183569
An exact proof term for the current goal is (mul_SNo_pos_pos eps0 (eps_ 1) Heps0S Heps1S Heps0PosS Heps1PosS).
L183569
We prove the intermediate claim HetaPos: Rlt 0 eta.
L183571
An exact proof term for the current goal is (RltI 0 eta real_0 HetaR HetaPosS).
L183571
We prove the intermediate claim HexK: ∃K ∈ ω, eps_ K < eta.
L183573
An exact proof term for the current goal is (exists_eps_lt_pos_Euclid eta HetaR HetaPos).
L183573
Apply HexK to the current goal.
L183574
Let K be given.
L183575
Assume HK.
L183575
We prove the intermediate claim HKomega: K ∈ ω.
L183577
An exact proof term for the current goal is (andEL (K ∈ ω) (eps_ K < eta) HK).
L183577
We prove the intermediate claim HKnat: nat_p K.
L183579
An exact proof term for the current goal is (omega_nat_p K HKomega).
L183579
We prove the intermediate claim HepsKltEtaS: eps_ K < eta.
L183581
An exact proof term for the current goal is (andER (K ∈ ω) (eps_ K < eta) HK).
L183581
We prove the intermediate claim HepsKR: eps_ K ∈ R.
L183583
An exact proof term for the current goal is (SNoS_omega_real (eps_ K) (SNo_eps_SNoS_omega K HKomega)).
L183583
We prove the intermediate claim HepsKS: SNo (eps_ K).
L183585
An exact proof term for the current goal is (real_SNo (eps_ K) HepsKR).
L183585
Set N0 to be the term add_nat K K.
L183586
We prove the intermediate claim HN0O: N0 ∈ ω.
L183588
rewrite the current goal using (add_nat_add_SNo K HKomega K HKomega) (from left to right).
L183588
An exact proof term for the current goal is (add_SNo_In_omega K HKomega K HKomega).
L183589
We prove the intermediate claim HN0Nat: nat_p N0.
L183591
An exact proof term for the current goal is (omega_nat_p N0 HN0O).
L183591
We prove the intermediate claim HdenS: SNo den.
L183593
An exact proof term for the current goal is (real_SNo den HdenR).
L183593
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
L183595
We prove the intermediate claim HdenLt1: den < 1.
L183597
rewrite the current goal using HdenDef (from left to right).
L183597
An exact proof term for the current goal is (RltE_lt two_thirds 1 Rlt_two_thirds_1).
L183598
Set den2 to be the term mul_SNo den den.
L183599
We prove the intermediate claim Hden2S: SNo den2.
L183601
An exact proof term for the current goal is (SNo_mul_SNo den den HdenS HdenS).
L183601
We prove the intermediate claim Hden2Lt_eps1: den2 < eps_ 1.
L183603
rewrite the current goal using HdenDef (from left to right) at position 1.
L183603
rewrite the current goal using HdenDef (from left to right) at position 2.
L183604
An exact proof term for the current goal is two_thirds_sq_lt_eps_1.
L183605
We prove the intermediate claim Hc_step: ∀m : set, nat_p m → ((State (ordsucc m)) 1) 1 = mul_SNo (((State m) 1) 1) den.
L183608
Let m be given.
L183608
Assume HmNat: nat_p m.
L183608
We prove the intermediate claim HS: State (ordsucc m) = StepState m (State m).
L183610
An exact proof term for the current goal is (nat_primrec_S BaseState StepState m HmNat).
L183610
rewrite the current goal using HS (from left to right).
L183611
Set st to be the term State m.
L183612
Set c to be the term (st 1) 1.
L183613
Set cNew to be the term mul_SNo c den.
L183614
We prove the intermediate claim Hdef: StepState m st = (compose_fun X (pair_map X (st 0) (compose_fun X (u_of ((st 1) 0)) (mul_const_fun c))) add_fun_R,(compose_fun A (compose_fun A (pair_map A ((st 1) 0) (compose_fun A (u_of ((st 1) 0)) neg_fun)) add_fun_R) (div_const_fun den),cNew)).
Use reflexivity.
L183627
We prove the intermediate claim HtEq: ((StepState m st) 1) 1 = cNew.
L183629
We prove the intermediate claim Hinner: (StepState m st) 1 = (compose_fun A (compose_fun A (pair_map A ((st 1) 0) (compose_fun A (u_of ((st 1) 0)) neg_fun)) add_fun_R) (div_const_fun den),cNew).
L183636
rewrite the current goal using Hdef (from left to right).
L183636
An exact proof term for the current goal is (tuple_2_1_eq (compose_fun X (pair_map X (st 0) (compose_fun X (u_of ((st 1) 0)) (mul_const_fun c))) add_fun_R) (compose_fun A (compose_fun A (pair_map A ((st 1) 0) (compose_fun A (u_of ((st 1) 0)) neg_fun)) add_fun_R) (div_const_fun den),cNew)).
L183648
rewrite the current goal using Hinner (from left to right).
L183649
An exact proof term for the current goal is (tuple_2_1_eq (compose_fun A (compose_fun A (pair_map A ((st 1) 0) (compose_fun A (u_of ((st 1) 0)) neg_fun)) add_fun_R) (div_const_fun den)) cNew).
L183656
rewrite the current goal using HtEq (from left to right).
L183657
We prove the intermediate claim HstEq: st = State m.
Use reflexivity.
L183659
We prove the intermediate claim HcEq2: c = ((State m) 1) 1.
L183661
rewrite the current goal using HstEq (from left to right).
Use reflexivity.
L183662
rewrite the current goal using HcEq2 (from left to right).
Use reflexivity.
L183664
We prove the intermediate claim Hc_even_lt: ∀t : set, nat_p t → ((State (add_nat t t)) 1) 1 < eps_ t.
L183667
Apply nat_ind to the current goal.
L183668
We will prove ((State (add_nat 0 0)) 1) 1 < eps_ 0.
L183668
rewrite the current goal using (add_nat_0R 0) (from left to right) at position 1.
L183669
rewrite the current goal using eps_0_1 (from left to right).
L183670
An exact proof term for the current goal is (HInv_c_lt1 0 (nat_p_omega 0 nat_0)).
L183672
Let t be given.
L183672
Assume HtNat: nat_p t.
L183672
Assume IH: ((State (add_nat t t)) 1) 1 < eps_ t.
L183673
We will prove ((State (add_nat (ordsucc t) (ordsucc t))) 1) 1 < eps_ (ordsucc t).
L183674
Set N to be the term add_nat t t.
L183675
We prove the intermediate claim HNnat: nat_p N.
L183677
An exact proof term for the current goal is (add_nat_p t HtNat t HtNat).
L183677
We prove the intermediate claim Hidx: add_nat (ordsucc t) (ordsucc t) = ordsucc (ordsucc N).
L183679
rewrite the current goal using (add_nat_SL t HtNat (ordsucc t) (nat_ordsucc t HtNat)) (from left to right).
L183679
rewrite the current goal using (add_nat_SR t t HtNat) (from left to right).
Use reflexivity.
L183681
rewrite the current goal using Hidx (from left to right).
L183682
We prove the intermediate claim HN0: N ∈ ω.
L183684
An exact proof term for the current goal is (nat_p_omega N HNnat).
L183684
We prove the intermediate claim HcSN: ((State (ordsucc N)) 1) 1 = mul_SNo (((State N) 1) 1) den.
L183686
An exact proof term for the current goal is (Hc_step N HNnat).
L183686
We prove the intermediate claim HcSSN: ((State (ordsucc (ordsucc N))) 1) 1 = mul_SNo (mul_SNo (((State N) 1) 1) den) den.
L183689
We prove the intermediate claim HSNnat: nat_p (ordsucc N).
L183690
An exact proof term for the current goal is (nat_ordsucc N HNnat).
L183690
rewrite the current goal using (Hc_step (ordsucc N) HSNnat) (from left to right).
L183691
rewrite the current goal using HcSN (from left to right).
Use reflexivity.
L183693
rewrite the current goal using HcSSN (from left to right).
L183694
rewrite the current goal using (mul_SNo_assoc (((State N) 1) 1) den den (real_SNo (((State N) 1) 1) (andEL (((State N) 1) 1 ∈ R) (0 < ((State N) 1) 1) (HInv_cpos N HN0))) HdenS HdenS) (from right to left).
L183697
We prove the intermediate claim Hden2Def: den2 = mul_SNo den den.
Use reflexivity.
L183699
rewrite the current goal using Hden2Def (from right to left).
L183700
We prove the intermediate claim HcNpos: 0 < ((State N) 1) 1.
L183702
An exact proof term for the current goal is (andER (((State N) 1) 1 ∈ R) (0 < ((State N) 1) 1) (HInv_cpos N HN0)).
L183702
We prove the intermediate claim HcNS: SNo (((State N) 1) 1).
L183704
An exact proof term for the current goal is (real_SNo (((State N) 1) 1) (andEL (((State N) 1) 1 ∈ R) (0 < ((State N) 1) 1) (HInv_cpos N HN0))).
L183704
rewrite the current goal using (mul_SNo_com (((State N) 1) 1) den2 HcNS Hden2S) (from left to right).
L183705
We prove the intermediate claim Heps1S0: SNo (eps_ 1).
L183707
An exact proof term for the current goal is (real_SNo (eps_ 1) eps_1_in_R).
L183707
We prove the intermediate claim Heps1pos: 0 < eps_ 1.
L183709
An exact proof term for the current goal is (RltE_lt 0 (eps_ 1) eps_1_pos_R).
L183709
We prove the intermediate claim Hmul1: mul_SNo den2 (((State N) 1) 1) < mul_SNo (eps_ 1) (((State N) 1) 1).
L183711
An exact proof term for the current goal is (pos_mul_SNo_Lt' den2 (eps_ 1) (((State N) 1) 1) Hden2S Heps1S0 HcNS HcNpos Hden2Lt_eps1).
L183712
We prove the intermediate claim Heps_tR: (eps_ t) ∈ R.
L183714
An exact proof term for the current goal is (SNoS_omega_real (eps_ t) (SNo_eps_SNoS_omega t (nat_p_omega t HtNat))).
L183714
We prove the intermediate claim Heps_tS: SNo (eps_ t).
L183716
An exact proof term for the current goal is (real_SNo (eps_ t) Heps_tR).
L183716
We prove the intermediate claim Hmul2: mul_SNo (eps_ 1) (((State N) 1) 1) < mul_SNo (eps_ 1) (eps_ t).
L183718
We prove the intermediate claim HcomL: mul_SNo (eps_ 1) (((State N) 1) 1) = mul_SNo (((State N) 1) 1) (eps_ 1).
L183719
An exact proof term for the current goal is (mul_SNo_com (eps_ 1) (((State N) 1) 1) Heps1S0 HcNS).
L183719
We prove the intermediate claim HcomR: mul_SNo (eps_ 1) (eps_ t) = mul_SNo (eps_ t) (eps_ 1).
L183721
An exact proof term for the current goal is (mul_SNo_com (eps_ 1) (eps_ t) Heps1S0 Heps_tS).
L183721
rewrite the current goal using HcomL (from left to right).
L183722
rewrite the current goal using HcomR (from left to right).
L183723
An exact proof term for the current goal is (pos_mul_SNo_Lt' (((State N) 1) 1) (eps_ t) (eps_ 1) HcNS Heps_tS Heps1S0 Heps1pos IH).
L183725
We prove the intermediate claim HmulTra: mul_SNo den2 (((State N) 1) 1) < mul_SNo (eps_ 1) (eps_ t).
L183727
An exact proof term for the current goal is (SNoLt_tra (mul_SNo den2 (((State N) 1) 1)) (mul_SNo (eps_ 1) (((State N) 1) 1)) (mul_SNo (eps_ 1) (eps_ t)) (SNo_mul_SNo den2 (((State N) 1) 1) Hden2S HcNS) (SNo_mul_SNo (eps_ 1) (((State N) 1) 1) Heps1S0 HcNS) (SNo_mul_SNo (eps_ 1) (eps_ t) Heps1S0 Heps_tS) Hmul1 Hmul2).
L183733
We prove the intermediate claim HepsEq: mul_SNo (eps_ 1) (eps_ t) = eps_ (ordsucc t).
L183735
rewrite the current goal using (mul_SNo_eps_eps_add_SNo 1 (nat_p_omega 1 nat_1) t (nat_p_omega t HtNat)) (from left to right).
L183735
We prove the intermediate claim Hordt: ordinal t.
L183737
An exact proof term for the current goal is (nat_p_ordinal t HtNat).
L183737
rewrite the current goal using (ordinal_ordsucc_SNo_eq t Hordt) (from right to left).
Use reflexivity.
L183739
rewrite the current goal using HepsEq (from right to left).
L183740
An exact proof term for the current goal is HmulTra.
L183741
We prove the intermediate claim HcN0: ((State N0) 1) 1 < eps_ K.
L183743
An exact proof term for the current goal is (Hc_even_lt K HKnat).
L183743
We prove the intermediate claim HcN0R: ((State N0) 1) 1 ∈ R.
L183745
An exact proof term for the current goal is (andEL (((State N0) 1) 1 ∈ R) (0 < ((State N0) 1) 1) (HInv_cpos N0 HN0O)).
L183745
We prove the intermediate claim HcN0S: SNo (((State N0) 1) 1).
L183747
An exact proof term for the current goal is (real_SNo (((State N0) 1) 1) HcN0R).
L183747
We prove the intermediate claim HcN0ltEta: ((State N0) 1) 1 < eta.
L183749
An exact proof term for the current goal is (SNoLt_tra (((State N0) 1) 1) (eps_ K) eta HcN0S HepsKS HetaS HcN0 HepsKltEtaS).
L183749
We use N0 to witness the existential quantifier.
L183750
Apply andI to the current goal.
L183752
An exact proof term for the current goal is HN0O.
L183753
Let m and n be given.
L183753
Assume HmO: m ∈ ω.
L183753
Assume HnO: n ∈ ω.
L183753
Assume HNm: N0 ⊆ m.
L183754
Assume HNn: N0 ⊆ n.
L183754
Let x be given.
L183755
Assume HxX: x ∈ X.
L183755
We will prove Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) eps0.
L183756
We prove the intermediate claim HmNat: nat_p m.
L183758
An exact proof term for the current goal is (omega_nat_p m HmO).
L183758
We prove the intermediate claim HnNat: nat_p n.
L183760
An exact proof term for the current goal is (omega_nat_p n HnO).
L183760
We prove the intermediate claim Hexkm: ∃km : set, nat_p km ∧ m = add_nat km N0.
L183762
An exact proof term for the current goal is (nat_Subq_add_ex N0 HN0Nat m HmNat HNm).
L183762
Apply Hexkm to the current goal.
L183763
Let km be given.
L183764
Assume Hkm.
L183764
We prove the intermediate claim HkmNat: nat_p km.
L183766
An exact proof term for the current goal is (andEL (nat_p km) (m = add_nat km N0) Hkm).
L183766
We prove the intermediate claim HmEq: m = add_nat km N0.
L183768
An exact proof term for the current goal is (andER (nat_p km) (m = add_nat km N0) Hkm).
L183768
rewrite the current goal using HmEq (from left to right).
L183769
We prove the intermediate claim Hexkn: ∃kn : set, nat_p kn ∧ n = add_nat kn N0.
L183771
An exact proof term for the current goal is (nat_Subq_add_ex N0 HN0Nat n HnNat HNn).
L183771
Apply Hexkn to the current goal.
L183772
Let kn be given.
L183773
Assume Hkn.
L183773
We prove the intermediate claim HknNat: nat_p kn.
L183775
An exact proof term for the current goal is (andEL (nat_p kn) (n = add_nat kn N0) Hkn).
L183775
We prove the intermediate claim HnEq: n = add_nat kn N0.
L183777
An exact proof term for the current goal is (andER (nat_p kn) (n = add_nat kn N0) Hkn).
L183777
rewrite the current goal using HnEq (from left to right).
L183778
We prove the intermediate claim HmO2: (add_nat km N0) ∈ ω.
L183780
An exact proof term for the current goal is (nat_p_omega (add_nat km N0) (add_nat_p km HkmNat N0 HN0Nat)).
L183780
We prove the intermediate claim HnO2: (add_nat kn N0) ∈ ω.
L183782
An exact proof term for the current goal is (nat_p_omega (add_nat kn N0) (add_nat_p kn HknNat N0 HN0Nat)).
L183782
We prove the intermediate claim HfnDef: fn = graph ω (λk : set ⇒ (State k) 0).
Use reflexivity.
L183784
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λk : set ⇒ (State k) 0) (add_nat km N0) HfnDef HmO2) (from left to right).
L183785
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λk : set ⇒ (State k) 0) (add_nat kn N0) HfnDef HnO2) (from left to right).
L183786
We prove the intermediate claim HmFS: (State (add_nat km N0)) 0 ∈ function_space X R.
L183788
An exact proof term for the current goal is (HInv_g_FS (add_nat km N0) HmO2).
L183788
We prove the intermediate claim HnFS: (State (add_nat kn N0)) 0 ∈ function_space X R.
L183790
An exact proof term for the current goal is (HInv_g_FS (add_nat kn N0) HnO2).
L183790
We prove the intermediate claim Hm_on: function_on ((State (add_nat km N0)) 0) X R.
L183792
An exact proof term for the current goal is (function_on_of_function_space ((State (add_nat km N0)) 0) X R HmFS).
L183792
We prove the intermediate claim Hn_on: function_on ((State (add_nat kn N0)) 0) X R.
L183794
An exact proof term for the current goal is (function_on_of_function_space ((State (add_nat kn N0)) 0) X R HnFS).
L183794
We prove the intermediate claim HmxR: apply_fun ((State (add_nat km N0)) 0) x ∈ R.
L183796
An exact proof term for the current goal is (Hm_on x HxX).
L183796
We prove the intermediate claim HnxR: apply_fun ((State (add_nat kn N0)) 0) x ∈ R.
L183798
An exact proof term for the current goal is (Hn_on x HxX).
L183798
We prove the intermediate claim HmxS: SNo (apply_fun ((State (add_nat km N0)) 0) x).
L183800
An exact proof term for the current goal is (real_SNo (apply_fun ((State (add_nat km N0)) 0) x) HmxR).
L183800
We prove the intermediate claim HnxS: SNo (apply_fun ((State (add_nat kn N0)) 0) x).
L183802
An exact proof term for the current goal is (real_SNo (apply_fun ((State (add_nat kn N0)) 0) x) HnxR).
L183802
We prove the intermediate claim HgN0FS: (State N0) 0 ∈ function_space X R.
L183804
An exact proof term for the current goal is (HInv_g_FS N0 HN0O).
L183804
We prove the intermediate claim HgN0_on: function_on ((State N0) 0) X R.
L183806
An exact proof term for the current goal is (function_on_of_function_space ((State N0) 0) X R HgN0FS).
L183806
We prove the intermediate claim HgN0xR: apply_fun ((State N0) 0) x ∈ R.
L183808
An exact proof term for the current goal is (HgN0_on x HxX).
L183808
We prove the intermediate claim HgN0xS: SNo (apply_fun ((State N0) 0) x).
L183810
An exact proof term for the current goal is (real_SNo (apply_fun ((State N0) 0) x) HgN0xR).
L183810
We prove the intermediate claim HcN0S0: SNo (((State N0) 1) 1).
L183812
An exact proof term for the current goal is (real_SNo (((State N0) 1) 1) HcN0R).
L183812
We prove the intermediate claim Hstep_budget: ∀t : set, nat_p t → ∀x0 : set, x0 ∈ X → abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x0) (minus_SNo (apply_fun ((State t) 0) x0))) ≤ add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1)).
(*** step bound for the g component ***)
L183818
Let t be given.
L183818
Assume HtNat: nat_p t.
L183818
Let x0 be given.
L183819
Assume Hx0X: x0 ∈ X.
L183819
We prove the intermediate claim HtO: t ∈ ω.
L183821
An exact proof term for the current goal is (nat_p_omega t HtNat).
L183821
We prove the intermediate claim HtSuccO: ordsucc t ∈ ω.
L183823
An exact proof term for the current goal is (omega_ordsucc t HtO).
L183823
Set r to be the term ((State t) 1) 0.
L183824
Set c to be the term ((State t) 1) 1.
L183825
Set corr to be the term compose_fun X (u_of r) (mul_const_fun c).
L183826
Set gNew to be the term compose_fun X (pair_map X ((State t) 0) corr) add_fun_R.
L183827
Set cNew to be the term mul_SNo c den.
L183828
We prove the intermediate claim HS: State (ordsucc t) = StepState t (State t).
L183830
An exact proof term for the current goal is (nat_primrec_S BaseState StepState t HtNat).
L183830
rewrite the current goal using (Hc_step t HtNat) (from left to right) at position 1.
L183831
We prove the intermediate claim HgSucc: (State (ordsucc t)) 0 = gNew.
L183833
rewrite the current goal using HS (from left to right).
L183833
We prove the intermediate claim HdefS: StepState t (State t) = (compose_fun X (pair_map X ((State t) 0) (compose_fun X (u_of (((State t) 1) 0)) (mul_const_fun (((State t) 1) 1)))) add_fun_R,(compose_fun A (compose_fun A (pair_map A (((State t) 1) 0) (compose_fun A (u_of (((State t) 1) 0)) neg_fun)) add_fun_R) (div_const_fun den),mul_SNo (((State t) 1) 1) den)).
Use reflexivity.
L183846
rewrite the current goal using HdefS (from left to right).
L183847
We prove the intermediate claim Hproj0: ((compose_fun X (pair_map X ((State t) 0) (compose_fun X (u_of (((State t) 1) 0)) (mul_const_fun (((State t) 1) 1)))) add_fun_R,(compose_fun A (compose_fun A (pair_map A (((State t) 1) 0) (compose_fun A (u_of (((State t) 1) 0)) neg_fun)) add_fun_R) (div_const_fun den),mul_SNo (((State t) 1) 1) den)) 0) = (compose_fun X (pair_map X ((State t) 0) (compose_fun X (u_of (((State t) 1) 0)) (mul_const_fun (((State t) 1) 1)))) add_fun_R).
L183866
An exact proof term for the current goal is (tuple_2_0_eq (compose_fun X (pair_map X ((State t) 0) (compose_fun X (u_of (((State t) 1) 0)) (mul_const_fun (((State t) 1) 1)))) add_fun_R) (compose_fun A (compose_fun A (pair_map A (((State t) 1) 0) (compose_fun A (u_of (((State t) 1) 0)) neg_fun)) add_fun_R) (div_const_fun den),mul_SNo (((State t) 1) 1) den)).
L183877
rewrite the current goal using Hproj0 (from left to right).
L183878
We prove the intermediate claim HrEq: r = ((State t) 1) 0.
Use reflexivity.
L183880
We prove the intermediate claim HcEq0: c = ((State t) 1) 1.
Use reflexivity.
L183882
rewrite the current goal using HrEq (from right to left).
L183883
rewrite the current goal using HcEq0 (from right to left).
L183884
We prove the intermediate claim HcorrDef: corr = compose_fun X (u_of r) (mul_const_fun c).
Use reflexivity.
L183886
rewrite the current goal using HcorrDef (from right to left).
L183887
We prove the intermediate claim HgNewDef: gNew = compose_fun X (pair_map X ((State t) 0) corr) add_fun_R.
Use reflexivity.
L183889
rewrite the current goal using HgNewDef (from left to right).
Use reflexivity.
L183891
We prove the intermediate claim HcEq: c = ((State t) 1) 1.
Use reflexivity.
L183893
We prove the intermediate claim HcR: c ∈ R.
L183895
rewrite the current goal using HcEq (from left to right).
L183895
An exact proof term for the current goal is (andEL (((State t) 1) 1 ∈ R) (0 < ((State t) 1) 1) (HInv_cpos t HtO)).
L183896
We prove the intermediate claim HcS: SNo c.
L183898
An exact proof term for the current goal is (real_SNo c HcR).
L183898
We prove the intermediate claim HcPos: 0 < c.
L183900
rewrite the current goal using HcEq (from left to right).
L183900
An exact proof term for the current goal is (andER (((State t) 1) 1 ∈ R) (0 < ((State t) 1) 1) (HInv_cpos t HtO)).
L183901
We prove the intermediate claim H0le_c: 0 ≤ c.
L183903
An exact proof term for the current goal is (SNoLtLe 0 c HcPos).
L183903
We prove the intermediate claim Hr_contI: continuous_map A Ta I Ti r.
L183905
An exact proof term for the current goal is (HInv_r_contI t HtO).
L183905
We prove the intermediate claim Hu_pack: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x1 : set, x1 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x1 = minus_SNo one_third) ∧ (∀x1 : set, x1 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x1 = one_third).
L183912
An exact proof term for the current goal is (Hu_of_prop r Hr_contI).
L183912
We prove the intermediate claim Hu_contI0: continuous_map X Tx I0 T0 (u_of r).
L183914
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r)) (∀x1 : set, x1 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x1 = minus_SNo one_third) (andEL ((continuous_map X Tx I0 T0 (u_of r)) ∧ (∀x1 : set, x1 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x1 = minus_SNo one_third)) (∀x1 : set, x1 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x1 = one_third) Hu_pack)).
L183922
We prove the intermediate claim Hu_fun: function_on (u_of r) X I0.
L183924
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 (u_of r) Hu_contI0).
L183924
We prove the intermediate claim HuxI0: apply_fun (u_of r) x0 ∈ I0.
L183926
An exact proof term for the current goal is (Hu_fun x0 Hx0X).
L183926
We prove the intermediate claim HuxR: apply_fun (u_of r) x0 ∈ R.
L183928
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun (u_of r) x0) HuxI0).
L183928
We prove the intermediate claim HuxS: SNo (apply_fun (u_of r) x0).
L183930
An exact proof term for the current goal is (real_SNo (apply_fun (u_of r) x0) HuxR).
L183930
We prove the intermediate claim HcorrEq: apply_fun corr x0 = mul_SNo (apply_fun (u_of r) x0) c.
L183933
We prove the intermediate claim HcorrDef0: corr = compose_fun X (u_of r) (mul_const_fun c).
Use reflexivity.
L183934
We prove the intermediate claim Hcomp: apply_fun corr x0 = apply_fun (mul_const_fun c) (apply_fun (u_of r) x0).
L183937
rewrite the current goal using HcorrDef0 (from left to right).
L183937
An exact proof term for the current goal is (compose_fun_apply X (u_of r) (mul_const_fun c) x0 Hx0X).
L183938
rewrite the current goal using Hcomp (from left to right).
L183939
An exact proof term for the current goal is (mul_const_fun_apply c (apply_fun (u_of r) x0) HcR HuxR).
L183940
We prove the intermediate claim HgxR: apply_fun ((State t) 0) x0 ∈ R.
L183942
An exact proof term for the current goal is (function_on_of_function_space ((State t) 0) X R (HInv_g_FS t HtO) x0 Hx0X).
L183942
We prove the intermediate claim HcorrR: apply_fun corr x0 ∈ R.
L183944
rewrite the current goal using HcorrEq (from left to right).
L183944
An exact proof term for the current goal is (real_mul_SNo (apply_fun (u_of r) x0) HuxR c HcR).
L183945
We prove the intermediate claim HgNewEval: apply_fun gNew x0 = add_SNo (apply_fun ((State t) 0) x0) (apply_fun corr x0).
L183948
An exact proof term for the current goal is (add_of_pair_map_apply X ((State t) 0) corr x0 Hx0X HgxR HcorrR).
L183948
rewrite the current goal using HgSucc (from left to right).
L183949
rewrite the current goal using HgNewEval (from left to right).
L183950
We prove the intermediate claim Ha0S: SNo (apply_fun ((State t) 0) x0).
L183952
An exact proof term for the current goal is (real_SNo (apply_fun ((State t) 0) x0) HgxR).
L183952
We prove the intermediate claim Hb0S: SNo (apply_fun corr x0).
L183954
An exact proof term for the current goal is (real_SNo (apply_fun corr x0) HcorrR).
L183954
We prove the intermediate claim Hcancel: add_SNo (add_SNo (apply_fun ((State t) 0) x0) (apply_fun corr x0)) (minus_SNo (apply_fun ((State t) 0) x0)) = apply_fun corr x0.
L183958
We prove the intermediate claim Hma0S: SNo (minus_SNo (apply_fun ((State t) 0) x0)).
L183959
An exact proof term for the current goal is (SNo_minus_SNo (apply_fun ((State t) 0) x0) Ha0S).
L183959
We prove the intermediate claim Hassoc1: add_SNo (apply_fun ((State t) 0) x0) (add_SNo (apply_fun corr x0) (minus_SNo (apply_fun ((State t) 0) x0))) = add_SNo (add_SNo (apply_fun ((State t) 0) x0) (apply_fun corr x0)) (minus_SNo (apply_fun ((State t) 0) x0)).
L183966
An exact proof term for the current goal is (add_SNo_assoc (apply_fun ((State t) 0) x0) (apply_fun corr x0) (minus_SNo (apply_fun ((State t) 0) x0)) Ha0S Hb0S Hma0S).
L183970
rewrite the current goal using Hassoc1 (from right to left) at position 1.
L183971
We prove the intermediate claim Hcom1: add_SNo (apply_fun corr x0) (minus_SNo (apply_fun ((State t) 0) x0)) = add_SNo (minus_SNo (apply_fun ((State t) 0) x0)) (apply_fun corr x0).
L183975
An exact proof term for the current goal is (add_SNo_com (apply_fun corr x0) (minus_SNo (apply_fun ((State t) 0) x0)) Hb0S Hma0S).
L183975
rewrite the current goal using Hcom1 (from left to right) at position 1.
L183976
We prove the intermediate claim Hassoc2: add_SNo (apply_fun ((State t) 0) x0) (add_SNo (minus_SNo (apply_fun ((State t) 0) x0)) (apply_fun corr x0)) = add_SNo (add_SNo (apply_fun ((State t) 0) x0) (minus_SNo (apply_fun ((State t) 0) x0))) (apply_fun corr x0).
L183983
An exact proof term for the current goal is (add_SNo_assoc (apply_fun ((State t) 0) x0) (minus_SNo (apply_fun ((State t) 0) x0)) (apply_fun corr x0) Ha0S Hma0S Hb0S).
L183987
rewrite the current goal using Hassoc2 (from left to right).
L183988
We prove the intermediate claim Hinv: add_SNo (apply_fun ((State t) 0) x0) (minus_SNo (apply_fun ((State t) 0) x0)) = 0.
L183990
An exact proof term for the current goal is (add_SNo_minus_SNo_rinv (apply_fun ((State t) 0) x0) Ha0S).
L183990
rewrite the current goal using Hinv (from left to right) at position 1.
L183991
An exact proof term for the current goal is (add_SNo_0L (apply_fun corr x0) Hb0S).
L183992
rewrite the current goal using Hcancel (from left to right) at position 1.
L183993
rewrite the current goal using HcorrEq (from left to right).
L183994
Set u to be the term apply_fun (u_of r) x0.
L183995
We prove the intermediate claim HuAbsLe: abs_SNo u ≤ one_third.
L183997
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L183998
An exact proof term for the current goal is (real_minus_SNo one_third one_third_in_R).
L183998
We prove the intermediate claim Hm13S: SNo (minus_SNo one_third).
L184000
An exact proof term for the current goal is (real_SNo (minus_SNo one_third) Hm13R).
L184000
We prove the intermediate claim H13S: SNo one_third.
L184002
An exact proof term for the current goal is (real_SNo one_third one_third_in_R).
L184002
We prove the intermediate claim Hbounds: Rle (minus_SNo one_third) u ∧ Rle u one_third.
L184004
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third u Hm13R one_third_in_R HuxI0).
L184004
We prove the intermediate claim Hlo: minus_SNo one_third ≤ u.
L184006
An exact proof term for the current goal is (SNoLe_of_Rle (minus_SNo one_third) u (andEL (Rle (minus_SNo one_third) u) (Rle u one_third) Hbounds)).
L184006
We prove the intermediate claim Hhi: u ≤ one_third.
L184008
An exact proof term for the current goal is (SNoLe_of_Rle u one_third (andER (Rle (minus_SNo one_third) u) (Rle u one_third) Hbounds)).
L184008
An exact proof term for the current goal is (abs_SNo_Le_of_bounds u one_third HuxS H13S Hlo Hhi).
L184009
We prove the intermediate claim Habsc: abs_SNo (mul_SNo u c) = mul_SNo (abs_SNo u) (abs_SNo c).
L184011
An exact proof term for the current goal is (abs_SNo_mul_eq u c HuxS HcS).
L184011
rewrite the current goal using Habsc (from left to right).
L184012
rewrite the current goal using (nonneg_abs_SNo c H0le_c) (from left to right).
L184013
We prove the intermediate claim H13S: SNo one_third.
L184015
An exact proof term for the current goal is (real_SNo one_third one_third_in_R).
L184015
We prove the intermediate claim HabsuS: SNo (abs_SNo u).
L184017
An exact proof term for the current goal is (SNo_abs_SNo u HuxS).
L184017
We prove the intermediate claim Hc13Le: mul_SNo (abs_SNo u) c ≤ mul_SNo one_third c.
L184019
An exact proof term for the current goal is (nonneg_mul_SNo_Le' (abs_SNo u) one_third c HabsuS H13S HcS H0le_c HuAbsLe).
L184019
We prove the intermediate claim HdenEq: den = two_thirds.
Use reflexivity.
L184021
rewrite the current goal using HdenEq (from left to right) at position 1.
L184022
We prove the intermediate claim H13Eq: mul_SNo one_third c = add_SNo c (minus_SNo (mul_SNo c two_thirds)).
L184024
We prove the intermediate claim H23S: SNo two_thirds.
L184025
An exact proof term for the current goal is (real_SNo two_thirds two_thirds_in_R).
L184025
We prove the intermediate claim Hc13S: SNo (mul_SNo c one_third).
L184027
An exact proof term for the current goal is (SNo_mul_SNo c one_third HcS H13S).
L184027
We prove the intermediate claim Hc23S: SNo (mul_SNo c two_thirds).
L184029
An exact proof term for the current goal is (SNo_mul_SNo c two_thirds HcS H23S).
L184029
rewrite the current goal using (mul_SNo_com one_third c H13S HcS) (from left to right).
L184030
We prove the intermediate claim HsumEq: add_SNo (mul_SNo c one_third) (mul_SNo c two_thirds) = c.
L184032
rewrite the current goal using (mul_SNo_distrL c one_third two_thirds HcS H13S H23S) (from right to left).
L184032
rewrite the current goal using add_one_third_two_thirds_eq_1 (from left to right) at position 1.
L184033
An exact proof term for the current goal is (mul_SNo_oneR c HcS).
L184034
We prove the intermediate claim Hcan: add_SNo (add_SNo (mul_SNo c one_third) (mul_SNo c two_thirds)) (minus_SNo (mul_SNo c two_thirds)) = mul_SNo c one_third.
L184037
An exact proof term for the current goal is (add_SNo_minus_R2 (mul_SNo c one_third) (mul_SNo c two_thirds) Hc13S Hc23S).
L184037
We prove the intermediate claim Htmp: add_SNo (add_SNo (mul_SNo c one_third) (mul_SNo c two_thirds)) (minus_SNo (mul_SNo c two_thirds)) = add_SNo c (minus_SNo (mul_SNo c two_thirds)).
L184040
rewrite the current goal using HsumEq (from left to right) at position 1.
Use reflexivity.
L184041
rewrite the current goal using Hcan (from right to left) at position 1.
L184042
rewrite the current goal using Htmp (from left to right) at position 1.
Use reflexivity.
L184044
rewrite the current goal using H13Eq (from right to left) at position 1.
L184045
An exact proof term for the current goal is Hc13Le.
L184046
We prove the intermediate claim Htail_budget: ∀k : set, nat_p k → ∀x0 : set, x0 ∈ X → abs_SNo (add_SNo (apply_fun ((State (add_nat k N0)) 0) x0) (minus_SNo (apply_fun ((State N0) 0) x0))) ≤ ((State N0) 1) 1.
(*** tail budget from N0 via nat induction on shift ***)
L184052
Let k be given.
L184052
Assume HkNat: nat_p k.
L184052
Let x0 be given.
L184053
Assume Hx0X: x0 ∈ X.
L184053
We prove the intermediate claim Htail_strong_all: ∀kk : set, nat_p kk → ∀x1 : set, x1 ∈ X → abs_SNo (add_SNo (apply_fun ((State (add_nat kk N0)) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))) ≤ add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat kk N0)) 1) 1)).
L184058
Apply nat_ind to the current goal.
L184059
We will prove ∀x1 : set, x1 ∈ X → abs_SNo (add_SNo (apply_fun ((State (add_nat 0 N0)) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))) ≤ add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat 0 N0)) 1) 1)).
L184061
Let x1 be given.
L184062
Assume Hx1X: x1 ∈ X.
L184062
We prove the intermediate claim Hadd0: add_nat 0 N0 = N0.
L184064
An exact proof term for the current goal is (add_nat_0L N0 HN0Nat).
L184064
rewrite the current goal using Hadd0 (from left to right) at position 1.
L184065
rewrite the current goal using Hadd0 (from left to right) at position 1.
L184066
We prove the intermediate claim HgN0FS: (State N0) 0 ∈ function_space X R.
L184068
An exact proof term for the current goal is (HInv_g_FS N0 HN0O).
L184068
We prove the intermediate claim HgN0_on: function_on ((State N0) 0) X R.
L184070
An exact proof term for the current goal is (function_on_of_function_space ((State N0) 0) X R HgN0FS).
L184070
We prove the intermediate claim HgN0xR: apply_fun ((State N0) 0) x1 ∈ R.
L184072
An exact proof term for the current goal is (HgN0_on x1 Hx1X).
L184072
We prove the intermediate claim HgN0xS: SNo (apply_fun ((State N0) 0) x1).
L184074
An exact proof term for the current goal is (real_SNo (apply_fun ((State N0) 0) x1) HgN0xR).
L184074
rewrite the current goal using (add_SNo_minus_SNo_rinv (apply_fun ((State N0) 0) x1) HgN0xS) (from left to right) at position 1.
L184075
rewrite the current goal using (nonneg_abs_SNo 0 (SNoLe_ref 0)) (from left to right) at position 1.
L184076
rewrite the current goal using (add_SNo_minus_SNo_rinv (((State N0) 1) 1) HcN0S0) (from left to right) at position 1.
L184077
An exact proof term for the current goal is (SNoLe_ref 0).
L184079
Let kk be given.
L184079
Assume HkkNat: nat_p kk.
L184079
Assume IH: ∀x1 : set, x1 ∈ X → abs_SNo (add_SNo (apply_fun ((State (add_nat kk N0)) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))) ≤ add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat kk N0)) 1) 1)).
L184082
We will prove ∀x1 : set, x1 ∈ X → abs_SNo (add_SNo (apply_fun ((State (add_nat (ordsucc kk) N0)) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))) ≤ add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat (ordsucc kk) N0)) 1) 1)).
L184085
Let x1 be given.
L184086
Assume Hx1X: x1 ∈ X.
L184086
Set t to be the term add_nat kk N0.
L184087
We prove the intermediate claim HtNat: nat_p t.
L184089
An exact proof term for the current goal is (add_nat_p kk HkkNat N0 HN0Nat).
L184089
We prove the intermediate claim HtO: t ∈ ω.
L184091
An exact proof term for the current goal is (nat_p_omega t HtNat).
L184091
We prove the intermediate claim Hidx: add_nat (ordsucc kk) N0 = ordsucc t.
L184093
rewrite the current goal using (add_nat_SL kk HkkNat N0 HN0Nat) (from left to right).
Use reflexivity.
L184094
rewrite the current goal using Hidx (from left to right) at position 1.
L184095
rewrite the current goal using Hidx (from left to right) at position 1.
L184096
We prove the intermediate claim HtSuccNat: nat_p (ordsucc t).
L184098
An exact proof term for the current goal is (nat_ordsucc t HtNat).
L184098
We prove the intermediate claim HtSuccO: ordsucc t ∈ ω.
L184100
An exact proof term for the current goal is (nat_p_omega (ordsucc t) HtSuccNat).
L184100
We prove the intermediate claim HgN0FS: (State N0) 0 ∈ function_space X R.
L184102
An exact proof term for the current goal is (HInv_g_FS N0 HN0O).
L184102
We prove the intermediate claim HgN0_on: function_on ((State N0) 0) X R.
L184104
An exact proof term for the current goal is (function_on_of_function_space ((State N0) 0) X R HgN0FS).
L184104
We prove the intermediate claim HgN0xR: apply_fun ((State N0) 0) x1 ∈ R.
L184106
An exact proof term for the current goal is (HgN0_on x1 Hx1X).
L184106
We prove the intermediate claim HgN0xS: SNo (apply_fun ((State N0) 0) x1).
L184108
An exact proof term for the current goal is (real_SNo (apply_fun ((State N0) 0) x1) HgN0xR).
L184108
We prove the intermediate claim HtFS: (State t) 0 ∈ function_space X R.
L184110
An exact proof term for the current goal is (HInv_g_FS t HtO).
L184110
We prove the intermediate claim Ht_on: function_on ((State t) 0) X R.
L184112
An exact proof term for the current goal is (function_on_of_function_space ((State t) 0) X R HtFS).
L184112
We prove the intermediate claim HtxR: apply_fun ((State t) 0) x1 ∈ R.
L184114
An exact proof term for the current goal is (Ht_on x1 Hx1X).
L184114
We prove the intermediate claim HtxS: SNo (apply_fun ((State t) 0) x1).
L184116
An exact proof term for the current goal is (real_SNo (apply_fun ((State t) 0) x1) HtxR).
L184116
We prove the intermediate claim HtSuccFS: (State (ordsucc t)) 0 ∈ function_space X R.
L184118
An exact proof term for the current goal is (HInv_g_FS (ordsucc t) HtSuccO).
L184118
We prove the intermediate claim HtSucc_on: function_on ((State (ordsucc t)) 0) X R.
L184120
An exact proof term for the current goal is (function_on_of_function_space ((State (ordsucc t)) 0) X R HtSuccFS).
L184120
We prove the intermediate claim HtSuccxR: apply_fun ((State (ordsucc t)) 0) x1 ∈ R.
L184122
An exact proof term for the current goal is (HtSucc_on x1 Hx1X).
L184122
We prove the intermediate claim HtSuccxS: SNo (apply_fun ((State (ordsucc t)) 0) x1).
L184124
An exact proof term for the current goal is (real_SNo (apply_fun ((State (ordsucc t)) 0) x1) HtSuccxR).
L184124
We prove the intermediate claim Htri: abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))) ≤ add_SNo (abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1)))) (abs_SNo (add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1)))).
L184130
An exact proof term for the current goal is (abs_SNo_triangle (apply_fun ((State (ordsucc t)) 0) x1) (apply_fun ((State t) 0) x1) (apply_fun ((State N0) 0) x1) HtSuccxS HtxS HgN0xS).
L184134
We prove the intermediate claim HstepLe: abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1))) ≤ add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1)).
L184138
An exact proof term for the current goal is (Hstep_budget t HtNat x1 Hx1X).
L184138
We prove the intermediate claim HihLe: abs_SNo (add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))) ≤ add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1)).
L184142
An exact proof term for the current goal is (IH x1 Hx1X).
L184142
We prove the intermediate claim Habs1S: SNo (abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1)))).
L184144
An exact proof term for the current goal is (SNo_abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1))) (SNo_add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1)) HtSuccxS (SNo_minus_SNo (apply_fun ((State t) 0) x1) HtxS))).
L184146
We prove the intermediate claim Habs2S: SNo (abs_SNo (add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1)))).
L184148
An exact proof term for the current goal is (SNo_abs_SNo (add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))) (SNo_add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1)) HtxS (SNo_minus_SNo (apply_fun ((State N0) 0) x1) HgN0xS))).
L184150
We prove the intermediate claim HctR: (((State t) 1) 1) ∈ R.
L184152
An exact proof term for the current goal is (andEL ((((State t) 1) 1 ∈ R)) (0 < (((State t) 1) 1)) (HInv_cpos t HtO)).
L184152
We prove the intermediate claim HctS: SNo (((State t) 1) 1).
L184154
An exact proof term for the current goal is (real_SNo (((State t) 1) 1) HctR).
L184154
We prove the intermediate claim HctsR: (((State (ordsucc t)) 1) 1) ∈ R.
L184156
An exact proof term for the current goal is (andEL ((((State (ordsucc t)) 1) 1 ∈ R)) (0 < (((State (ordsucc t)) 1) 1)) (HInv_cpos (ordsucc t) HtSuccO)).
L184156
We prove the intermediate claim HctsS: SNo (((State (ordsucc t)) 1) 1).
L184158
An exact proof term for the current goal is (real_SNo (((State (ordsucc t)) 1) 1) HctsR).
L184158
We prove the intermediate claim Hc0S: SNo (((State N0) 1) 1).
L184160
An exact proof term for the current goal is HcN0S0.
L184160
We prove the intermediate claim HmctS: SNo (minus_SNo (((State t) 1) 1)).
L184162
An exact proof term for the current goal is (SNo_minus_SNo (((State t) 1) 1) HctS).
L184162
We prove the intermediate claim HmctsS: SNo (minus_SNo (((State (ordsucc t)) 1) 1)).
L184164
An exact proof term for the current goal is (SNo_minus_SNo (((State (ordsucc t)) 1) 1) HctsS).
L184164
We prove the intermediate claim Hrhs1S: SNo (add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1))).
L184166
An exact proof term for the current goal is (SNo_add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1)) HctS HmctsS).
L184166
We prove the intermediate claim Hrhs2S: SNo (add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1))).
L184168
An exact proof term for the current goal is (SNo_add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1)) Hc0S HmctS).
L184168
We prove the intermediate claim HsumLe: add_SNo (abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1)))) (abs_SNo (add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1)))) ≤ add_SNo (add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1))) (add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1))).
L184175
An exact proof term for the current goal is (add_SNo_Le3 (abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1)))) (abs_SNo (add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1)))) (add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1))) (add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1))) Habs1S Habs2S Hrhs1S Hrhs2S HstepLe HihLe).
L184180
We prove the intermediate claim HsumEq: add_SNo (add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1))) (add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1))) = add_SNo (((State N0) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1)).
L184186
rewrite the current goal using (add_SNo_com (add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1))) (add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1))) Hrhs1S Hrhs2S) (from left to right) at position 1.
L184189
rewrite the current goal using (add_SNo_com_4_inner_mid (((State N0) 1) 1) (minus_SNo (((State t) 1) 1)) (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1)) Hc0S HmctS HctS HmctsS) (from left to right) at position 1.
L184195
We prove the intermediate claim Ht2S: SNo (add_SNo (((State N0) 1) 1) (((State t) 1) 1)).
L184197
An exact proof term for the current goal is (SNo_add_SNo (((State N0) 1) 1) (((State t) 1) 1) Hc0S HctS).
L184197
We prove the intermediate claim Ht3S: SNo (add_SNo (minus_SNo (((State t) 1) 1)) (minus_SNo (((State (ordsucc t)) 1) 1))).
L184199
An exact proof term for the current goal is (SNo_add_SNo (minus_SNo (((State t) 1) 1)) (minus_SNo (((State (ordsucc t)) 1) 1)) HmctS HmctsS).
L184199
rewrite the current goal using (add_SNo_assoc (((State N0) 1) 1) (((State t) 1) 1) (add_SNo (minus_SNo (((State t) 1) 1)) (minus_SNo (((State (ordsucc t)) 1) 1))) Hc0S HctS Ht3S) (from right to left) at position 1.
L184202
rewrite the current goal using (add_SNo_assoc (((State t) 1) 1) (minus_SNo (((State t) 1) 1)) (minus_SNo (((State (ordsucc t)) 1) 1)) HctS HmctS HmctsS) (from left to right) at position 1.
L184204
We prove the intermediate claim Hinv: add_SNo (((State t) 1) 1) (minus_SNo (((State t) 1) 1)) = 0.
L184206
An exact proof term for the current goal is (add_SNo_minus_SNo_rinv (((State t) 1) 1) HctS).
L184206
rewrite the current goal using Hinv (from left to right) at position 1.
L184207
rewrite the current goal using (add_SNo_0L (minus_SNo (((State (ordsucc t)) 1) 1)) HmctsS) (from left to right) at position 1.
Use reflexivity.
L184209
rewrite the current goal using HsumEq (from right to left) at position 1.
L184210
An exact proof term for the current goal is (SNoLe_tra (abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1)))) (add_SNo (abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1)))) (abs_SNo (add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))))) (add_SNo (add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1))) (add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1)))) (SNo_abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))) (SNo_add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1)) HtSuccxS (SNo_minus_SNo (apply_fun ((State N0) 0) x1) HgN0xS))) (SNo_add_SNo (abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1)))) (abs_SNo (add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1)))) Habs1S Habs2S) (SNo_add_SNo (add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1))) (add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1))) Hrhs1S Hrhs2S) Htri HsumLe).
L184229
We prove the intermediate claim Hstrong: abs_SNo (add_SNo (apply_fun ((State (add_nat k N0)) 0) x0) (minus_SNo (apply_fun ((State N0) 0) x0))) ≤ add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat k N0)) 1) 1)).
L184233
An exact proof term for the current goal is (Htail_strong_all k HkNat x0 Hx0X).
L184233
We prove the intermediate claim HkN0Nat: nat_p (add_nat k N0).
L184235
An exact proof term for the current goal is (add_nat_p k HkNat N0 HN0Nat).
L184235
We prove the intermediate claim HkN0O: (add_nat k N0) ∈ ω.
L184237
An exact proof term for the current goal is (nat_p_omega (add_nat k N0) HkN0Nat).
L184237
We prove the intermediate claim HckR: (((State (add_nat k N0)) 1) 1) ∈ R.
L184239
An exact proof term for the current goal is (andEL ((((State (add_nat k N0)) 1) 1 ∈ R)) (0 < (((State (add_nat k N0)) 1) 1)) (HInv_cpos (add_nat k N0) HkN0O)).
L184239
We prove the intermediate claim HckS: SNo (((State (add_nat k N0)) 1) 1).
L184241
An exact proof term for the current goal is (real_SNo (((State (add_nat k N0)) 1) 1) HckR).
L184241
We prove the intermediate claim HckPos: 0 < (((State (add_nat k N0)) 1) 1).
L184243
An exact proof term for the current goal is (andER ((((State (add_nat k N0)) 1) 1 ∈ R)) (0 < (((State (add_nat k N0)) 1) 1)) (HInv_cpos (add_nat k N0) HkN0O)).
L184243
We prove the intermediate claim H0le_ck: 0 ≤ (((State (add_nat k N0)) 1) 1).
L184245
An exact proof term for the current goal is (SNoLtLe 0 (((State (add_nat k N0)) 1) 1) HckPos).
L184245
We prove the intermediate claim HmckLe0: minus_SNo (((State (add_nat k N0)) 1) 1) ≤ 0.
L184247
We prove the intermediate claim Htmp: minus_SNo (((State (add_nat k N0)) 1) 1) ≤ minus_SNo 0.
L184248
An exact proof term for the current goal is (minus_SNo_Le_contra 0 (((State (add_nat k N0)) 1) 1) SNo_0 HckS H0le_ck).
L184248
We prove the intermediate claim Hm0le0: minus_SNo 0 ≤ 0.
L184250
rewrite the current goal using minus_SNo_0 (from left to right).
L184250
An exact proof term for the current goal is (SNoLe_ref 0).
L184251
An exact proof term for the current goal is (SNoLe_tra (minus_SNo (((State (add_nat k N0)) 1) 1)) (minus_SNo 0) 0 (SNo_minus_SNo (((State (add_nat k N0)) 1) 1) HckS) (SNo_minus_SNo 0 SNo_0) SNo_0 Htmp Hm0le0).
L184259
We prove the intermediate claim HrhsLe: add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat k N0)) 1) 1)) ≤ ((State N0) 1) 1.
L184262
We prove the intermediate claim HsumLe0: add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat k N0)) 1) 1)) ≤ add_SNo (((State N0) 1) 1) 0.
L184265
An exact proof term for the current goal is (add_SNo_Le2 (((State N0) 1) 1) (minus_SNo (((State (add_nat k N0)) 1) 1)) 0 HcN0S0 (SNo_minus_SNo (((State (add_nat k N0)) 1) 1) HckS) SNo_0 HmckLe0).
L184265
We prove the intermediate claim HmidLe: add_SNo (((State N0) 1) 1) 0 ≤ ((State N0) 1) 1.
L184268
rewrite the current goal using (add_SNo_0R (((State N0) 1) 1) HcN0S0) (from left to right).
L184268
An exact proof term for the current goal is (SNoLe_ref (((State N0) 1) 1)).
L184269
An exact proof term for the current goal is (SNoLe_tra (add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat k N0)) 1) 1))) (add_SNo (((State N0) 1) 1) 0) (((State N0) 1) 1) (SNo_add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat k N0)) 1) 1)) HcN0S0 (SNo_minus_SNo (((State (add_nat k N0)) 1) 1) HckS)) (SNo_add_SNo (((State N0) 1) 1) 0 HcN0S0 SNo_0) HcN0S0 HsumLe0 HmidLe).
L184277
We prove the intermediate claim HgN0xS: SNo (apply_fun ((State N0) 0) x0).
L184279
We prove the intermediate claim HgN0FS: (State N0) 0 ∈ function_space X R.
L184280
An exact proof term for the current goal is (HInv_g_FS N0 HN0O).
L184280
We prove the intermediate claim HgN0_on: function_on ((State N0) 0) X R.
L184282
An exact proof term for the current goal is (function_on_of_function_space ((State N0) 0) X R HgN0FS).
L184282
We prove the intermediate claim HgN0xR: apply_fun ((State N0) 0) x0 ∈ R.
L184284
An exact proof term for the current goal is (HgN0_on x0 Hx0X).
L184284
An exact proof term for the current goal is (real_SNo (apply_fun ((State N0) 0) x0) HgN0xR).
L184285
An exact proof term for the current goal is (SNoLe_tra (abs_SNo (add_SNo (apply_fun ((State (add_nat k N0)) 0) x0) (minus_SNo (apply_fun ((State N0) 0) x0)))) (add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat k N0)) 1) 1))) (((State N0) 1) 1) (SNo_abs_SNo (add_SNo (apply_fun ((State (add_nat k N0)) 0) x0) (minus_SNo (apply_fun ((State N0) 0) x0))) (SNo_add_SNo (apply_fun ((State (add_nat k N0)) 0) x0) (minus_SNo (apply_fun ((State N0) 0) x0)) (real_SNo (apply_fun ((State (add_nat k N0)) 0) x0) (function_on_of_function_space ((State (add_nat k N0)) 0) X R (HInv_g_FS (add_nat k N0) HkN0O) x0 Hx0X)) (SNo_minus_SNo (apply_fun ((State N0) 0) x0) HgN0xS))) (SNo_add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat k N0)) 1) 1)) HcN0S0 (SNo_minus_SNo (((State (add_nat k N0)) 1) 1) HckS)) HcN0S0 Hstrong HrhsLe).
L184297
We prove the intermediate claim Habs_mN0: abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x))) < eta.
(*** derive abs difference < eps0 and conclude in bounded distance ***)
L184300
An exact proof term for the current goal is (SNoLeLt_tra (abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)))) (((State N0) 1) 1) eta (SNo_abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x))) (SNo_add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)) HmxS (SNo_minus_SNo (apply_fun ((State N0) 0) x) HgN0xS))) HcN0S0 HetaS (Htail_budget km HkmNat x HxX) HcN0ltEta).
L184307
We prove the intermediate claim HabSwap: abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x))) = abs_SNo (add_SNo (apply_fun ((State (add_nat kn N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x))).
L184310
An exact proof term for the current goal is (abs_SNo_dist_swap (apply_fun ((State N0) 0) x) (apply_fun ((State (add_nat kn N0)) 0) x) HgN0xS HnxS).
L184310
We prove the intermediate claim Habs_nN0: abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x))) < eta.
L184312
rewrite the current goal using HabSwap (from left to right).
L184312
An exact proof term for the current goal is (SNoLeLt_tra (abs_SNo (add_SNo (apply_fun ((State (add_nat kn N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)))) (((State N0) 1) 1) eta (SNo_abs_SNo (add_SNo (apply_fun ((State (add_nat kn N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x))) (SNo_add_SNo (apply_fun ((State (add_nat kn N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)) HnxS (SNo_minus_SNo (apply_fun ((State N0) 0) x) HgN0xS))) HcN0S0 HetaS (Htail_budget kn HknNat x HxX) HcN0ltEta).
L184320
We prove the intermediate claim HetaSumEq: add_SNo eta eta = eps0.
L184322
rewrite the current goal using (mul_SNo_distrL eps0 (eps_ 1) (eps_ 1) Heps0S Heps1S Heps1S) (from right to left).
L184322
rewrite the current goal using (eps_ordsucc_half_add 0 nat_0) (from left to right).
L184323
rewrite the current goal using eps_0_1 (from left to right).
L184324
rewrite the current goal using (mul_SNo_oneR eps0 Heps0S) (from left to right).
Use reflexivity.
L184326
We prove the intermediate claim HsumLt: add_SNo (abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)))) (abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x)))) < eps0.
L184331
rewrite the current goal using HetaSumEq (from right to left).
L184331
An exact proof term for the current goal is (add_SNo_Lt3 (abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)))) (abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x)))) eta eta (SNo_abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x))) (SNo_add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)) HmxS (SNo_minus_SNo (apply_fun ((State N0) 0) x) HgN0xS))) (SNo_abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x))) (SNo_add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x)) HgN0xS (SNo_minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x) HnxS))) HetaS HetaS Habs_mN0 Habs_nN0).
L184341
We prove the intermediate claim HtriMN: abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x))) ≤ add_SNo (abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)))) (abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x)))).
L184346
An exact proof term for the current goal is (abs_SNo_triangle (apply_fun ((State (add_nat km N0)) 0) x) (apply_fun ((State N0) 0) x) (apply_fun ((State (add_nat kn N0)) 0) x) HmxS HgN0xS HnxS).
L184347
We prove the intermediate claim HabsLt: abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x))) < eps0.
L184349
An exact proof term for the current goal is (SNoLeLt_tra (abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x)))) (add_SNo (abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)))) (abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x))))) eps0 (SNo_abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x))) (SNo_add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x)) HmxS (SNo_minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x) HnxS))) (SNo_add_SNo (abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)))) (abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x)))) (SNo_abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x))) (SNo_add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)) HmxS (SNo_minus_SNo (apply_fun ((State N0) 0) x) HgN0xS))) (SNo_abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x))) (SNo_add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x)) HgN0xS (SNo_minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x) HnxS)))) Heps0S HtriMN HsumLt).
L184367
We prove the intermediate claim Hpair: (apply_fun ((State (add_nat km N0)) 0) x,apply_fun ((State (add_nat kn N0)) 0) x) ∈ setprod R R.
L184369
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun ((State (add_nat km N0)) 0) x) (apply_fun ((State (add_nat kn N0)) 0) x) HmxR HnxR).
L184369
We prove the intermediate claim HdefM: R_bounded_metric = graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)).
Use reflexivity.
L184372
rewrite the current goal using HdefM (from left to right).
L184373
rewrite the current goal using (apply_fun_graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)) (apply_fun ((State (add_nat km N0)) 0) x,apply_fun ((State (add_nat kn N0)) 0) x) Hpair) (from left to right).
L184375
rewrite the current goal using (tuple_2_0_eq (apply_fun ((State (add_nat km N0)) 0) x) (apply_fun ((State (add_nat kn N0)) 0) x)) (from left to right).
L184376
rewrite the current goal using (tuple_2_1_eq (apply_fun ((State (add_nat km N0)) 0) x) (apply_fun ((State (add_nat kn N0)) 0) x)) (from left to right).
L184377
An exact proof term for the current goal is (abs_lt_lt1_imp_R_bounded_distance_lt (apply_fun ((State (add_nat km N0)) 0) x) (apply_fun ((State (add_nat kn N0)) 0) x) eps0 HmxR HnxR Heps0R Heps0Lt1 HabsLt).
L184382
Apply (SNoLt_trichotomy_or_impred eps 1 HepsS SNo_1 (∃N : set, N ∈ ω ∧ ∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) eps)) to the current goal.
L184388
Assume HepsLt1S: eps < 1.
L184388
We prove the intermediate claim HepsLt1: Rlt eps 1.
L184390
An exact proof term for the current goal is (RltI eps 1 HepsR real_1 HepsLt1S).
L184390
An exact proof term for the current goal is (Huc_small eps HepsR HepsPos HepsLt1).
L184392
Assume HepsEq1: eps = 1.
L184392
We prove the intermediate claim HexN: ∃N : set, N ∈ ω ∧ ∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) (eps_ 1).
L184397
An exact proof term for the current goal is (Huc_small (eps_ 1) eps_1_in_R eps_1_pos_R eps_1_lt1_R).
L184397
Apply HexN to the current goal.
L184398
Let N be given.
L184399
Assume HN.
L184399
We use N to witness the existential quantifier.
L184400
Apply andI to the current goal.
L184402
An exact proof term for the current goal is (andEL (N ∈ ω) (∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) (eps_ 1)) HN).
L184407
Let m and n be given.
L184407
Assume HmO: m ∈ ω.
L184407
Assume HnO: n ∈ ω.
L184407
Assume HNm: N ⊆ m.
L184408
Assume HNn: N ⊆ n.
L184408
Let x be given.
L184409
Assume HxX: x ∈ X.
L184409
We prove the intermediate claim HNprop: ∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) (eps_ 1).
L184414
An exact proof term for the current goal is (andER (N ∈ ω) (∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) (eps_ 1)) HN).
L184418
We prove the intermediate claim Hlt1: Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) (eps_ 1).
L184420
An exact proof term for the current goal is (HNprop m n HmO HnO HNm HNn x HxX).
L184420
We prove the intermediate claim Hlt2: Rlt (eps_ 1) eps.
L184422
rewrite the current goal using HepsEq1 (from left to right).
L184422
An exact proof term for the current goal is eps_1_lt1_R.
L184423
An exact proof term for the current goal is (Rlt_tra (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) (eps_ 1) eps Hlt1 Hlt2).
L184425
Assume H1LtEpsS: 1 < eps.
L184425
We prove the intermediate claim H1LtEps: Rlt 1 eps.
L184427
An exact proof term for the current goal is (RltI 1 eps real_1 HepsR H1LtEpsS).
L184427
We use 0 to witness the existential quantifier.
L184428
Apply andI to the current goal.
L184430
An exact proof term for the current goal is (nat_p_omega 0 nat_0).
L184431
Let m and n be given.
L184431
Assume HmO: m ∈ ω.
L184431
Assume HnO: n ∈ ω.
L184431
Assume HNm: 0 ⊆ m.
L184432
Assume HNn: 0 ⊆ n.
L184432
Let x be given.
L184433
Assume HxX: x ∈ X.
L184433
We will prove Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) eps.
L184434
We prove the intermediate claim HfnDef: fn = graph ω (λk : set ⇒ (State k) 0).
Use reflexivity.
L184436
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λk : set ⇒ (State k) 0) m HfnDef HmO) (from left to right).
L184437
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λk : set ⇒ (State k) 0) n HfnDef HnO) (from left to right).
L184438
We prove the intermediate claim HmFS: (State m) 0 ∈ function_space X R.
L184440
An exact proof term for the current goal is (HInv_g_FS m HmO).
L184440
We prove the intermediate claim HnFS: (State n) 0 ∈ function_space X R.
L184442
An exact proof term for the current goal is (HInv_g_FS n HnO).
L184442
We prove the intermediate claim Hm_on: function_on ((State m) 0) X R.
L184444
An exact proof term for the current goal is (function_on_of_function_space ((State m) 0) X R HmFS).
L184444
We prove the intermediate claim Hn_on: function_on ((State n) 0) X R.
L184446
An exact proof term for the current goal is (function_on_of_function_space ((State n) 0) X R HnFS).
L184446
We prove the intermediate claim HmxR: apply_fun ((State m) 0) x ∈ R.
L184448
An exact proof term for the current goal is (Hm_on x HxX).
L184448
We prove the intermediate claim HnxR: apply_fun ((State n) 0) x ∈ R.
L184450
An exact proof term for the current goal is (Hn_on x HxX).
L184450
We prove the intermediate claim Hpair: (apply_fun ((State m) 0) x,apply_fun ((State n) 0) x) ∈ setprod R R.
L184452
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun ((State m) 0) x) (apply_fun ((State n) 0) x) HmxR HnxR).
L184452
We prove the intermediate claim HdefM: R_bounded_metric = graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)).
Use reflexivity.
L184455
rewrite the current goal using HdefM (from left to right).
L184456
rewrite the current goal using (apply_fun_graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)) (apply_fun ((State m) 0) x,apply_fun ((State n) 0) x) Hpair) (from left to right).
L184458
rewrite the current goal using (tuple_2_0_eq (apply_fun ((State m) 0) x) (apply_fun ((State n) 0) x)) (from left to right).
L184459
rewrite the current goal using (tuple_2_1_eq (apply_fun ((State m) 0) x) (apply_fun ((State n) 0) x)) (from left to right).
L184460
We prove the intermediate claim Hle1: Rle (R_bounded_distance (apply_fun ((State m) 0) x) (apply_fun ((State n) 0) x)) 1.
L184462
An exact proof term for the current goal is (R_bounded_distance_le_1 (apply_fun ((State m) 0) x) (apply_fun ((State n) 0) x) HmxR HnxR).
L184462
An exact proof term for the current goal is (Rle_Rlt_tra (R_bounded_distance (apply_fun ((State m) 0) x) (apply_fun ((State n) 0) x)) 1 eps Hle1 H1LtEps).
L184463
Apply Hexfn to the current goal.
L184464
Let fn be given.
L184465
Assume Hfnpack: function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n)) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I) ∧ (∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x)) ∧ uniform_cauchy_metric X R R_bounded_metric fn.
L184473
We prove the intermediate claim Hfn1234: ((function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I)) ∧ (∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x)).
L184484
An exact proof term for the current goal is (andEL (((function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I)) ∧ (∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x))) (uniform_cauchy_metric X R R_bounded_metric fn) Hfnpack).
L184495
We prove the intermediate claim Hfn3: uniform_cauchy_metric X R R_bounded_metric fn.
L184497
An exact proof term for the current goal is (andER ((((function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I)) ∧ (∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x)))) (uniform_cauchy_metric X R R_bounded_metric fn) Hfnpack).
L184508
We prove the intermediate claim Hfn123: (function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I).
L184514
An exact proof term for the current goal is (andEL ((function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I)) (∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x)) Hfn1234).
L184523
We prove the intermediate claim HfnLimA: ∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x).
L184529
An exact proof term for the current goal is (andER ((function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I)) (∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x)) Hfn1234).
L184538
We prove the intermediate claim Hfn12: function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n)).
L184542
An exact proof term for the current goal is (andEL (function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I) Hfn123).
L184546
We prove the intermediate claim HfnRange: ∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I.
L184549
An exact proof term for the current goal is (andER (function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I) Hfn123).
L184553
We prove the intermediate claim Hfn1: function_on fn ω (function_space X R).
L184555
An exact proof term for the current goal is (andEL (function_on fn ω (function_space X R)) (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n)) Hfn12).
L184558
We prove the intermediate claim Hfn2: ∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n).
L184560
An exact proof term for the current goal is (andER (function_on fn ω (function_space X R)) (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n)) Hfn12).
L184563
We prove the intermediate claim HexgR: ∃gR : set, function_on gR X R ∧ uniform_limit_metric X R R_bounded_metric fn gR ∧ continuous_map X Tx R R_standard_topology gR.
L184568
An exact proof term for the current goal is (uniform_cauchy_continuous_to_R_has_continuous_limit X Tx fn HTx Hfn1 Hfn2 Hfn3).
L184573
Apply HexgR to the current goal.
L184574
Let gR be given.
L184575
Assume HgRpack: function_on gR X R ∧ uniform_limit_metric X R R_bounded_metric fn gR ∧ continuous_map X Tx R R_standard_topology gR.
L184578
We use gR to witness the existential quantifier.
L184579
Apply andI to the current goal.
L184581
Apply andI to the current goal.
L184582
An exact proof term for the current goal is (andER (function_on gR X R ∧ uniform_limit_metric X R R_bounded_metric fn gR) (continuous_map X Tx R R_standard_topology gR) HgRpack).
L184585
Let x be given.
L184585
Assume HxA: x ∈ A.
L184585
We will prove apply_fun gR x = apply_fun f x.
L184586
Apply (xm (apply_fun gR x = apply_fun f x)) to the current goal.
L184588
Assume Heq: apply_fun gR x = apply_fun f x.
L184588
An exact proof term for the current goal is Heq.
L184590
Assume Hneq: ¬ (apply_fun gR x = apply_fun f x).
L184590
We will prove apply_fun gR x = apply_fun f x.
L184591
Apply FalseE to the current goal.
L184592
We prove the intermediate claim HxX': x ∈ X.
L184594
An exact proof term for the current goal is (HAsubX x HxA).
L184594
We prove the intermediate claim HmR: metric_on R R_bounded_metric.
L184596
An exact proof term for the current goal is R_bounded_metric_is_metric_on_early.
L184596
We prove the intermediate claim HHaus: Hausdorff_space R (metric_topology R R_bounded_metric).
L184598
An exact proof term for the current goal is (metric_topology_Hausdorff R R_bounded_metric HmR).
L184598
We prove the intermediate claim HgR12: function_on gR X R ∧ uniform_limit_metric X R R_bounded_metric fn gR.
L184600
An exact proof term for the current goal is (andEL (function_on gR X R ∧ uniform_limit_metric X R R_bounded_metric fn gR) (continuous_map X Tx R R_standard_topology gR) HgRpack).
L184603
We prove the intermediate claim HgRfun: function_on gR X R.
L184605
An exact proof term for the current goal is (andEL (function_on gR X R) (uniform_limit_metric X R R_bounded_metric fn gR) HgR12).
L184608
We prove the intermediate claim HunifRgR: uniform_limit_metric X R R_bounded_metric fn gR.
L184610
An exact proof term for the current goal is (andER (function_on gR X R) (uniform_limit_metric X R R_bounded_metric fn gR) HgR12).
L184613
Set seqx to be the term graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x).
L184614
We prove the intermediate claim Hxlim_g: converges_to R (metric_topology R R_bounded_metric) seqx (apply_fun gR x).
L184616
An exact proof term for the current goal is (uniform_limit_metric_imp_converges_to_metric_topology_at_point X R R_bounded_metric fn gR x HmR Hfn1 HgRfun HxX' HunifRgR).
L184618
We prove the intermediate claim Hxlim_f: converges_to R (metric_topology R R_bounded_metric) seqx (apply_fun f x).
L184620
An exact proof term for the current goal is (HfnLimA x HxA).
L184620
We prove the intermediate claim HgxR: apply_fun gR x ∈ R.
L184622
An exact proof term for the current goal is (HgRfun x HxX').
L184622
We prove the intermediate claim HfxR: apply_fun f x ∈ R.
L184624
An exact proof term for the current goal is (Hf_R x HxA).
L184624
We prove the intermediate claim Hneqxy: apply_fun gR x ≠ apply_fun f x.
L184626
Assume Heq: apply_fun gR x = apply_fun f x.
L184626
An exact proof term for the current goal is (Hneq Heq).
L184627
We prove the intermediate claim Hseq_on: function_on seqx ω R.
L184629
Let n be given.
L184629
Assume HnO: n ∈ ω.
L184629
We will prove apply_fun seqx n ∈ R.
L184630
We prove the intermediate claim Hseqxdef: seqx = graph ω (λn0 : set ⇒ apply_fun (apply_fun fn n0) x).
Use reflexivity.
L184632
rewrite the current goal using (apply_fun_of_graph_eq seqx ω (λn0 : set ⇒ apply_fun (apply_fun fn n0) x) n Hseqxdef HnO) (from left to right).
L184633
We prove the intermediate claim HfnnxI: apply_fun (apply_fun fn n) x ∈ I.
L184635
An exact proof term for the current goal is (HfnRange n HnO x HxX').
L184635
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun (apply_fun fn n) x) HfnnxI).
L184636
We prove the intermediate claim Hnbhd_g: ∀U : set, U ∈ metric_topology R R_bounded_metric → apply_fun gR x ∈ U → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seqx n ∈ U.
L184640
An exact proof term for the current goal is (converges_to_neighborhoods R (metric_topology R R_bounded_metric) seqx (apply_fun gR x) Hxlim_g).
L184642
We prove the intermediate claim Hnbhd_f: ∀U : set, U ∈ metric_topology R R_bounded_metric → apply_fun f x ∈ U → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seqx n ∈ U.
L184646
An exact proof term for the current goal is (converges_to_neighborhoods R (metric_topology R R_bounded_metric) seqx (apply_fun f x) Hxlim_f).
L184648
An exact proof term for the current goal is (Hausdorff_unique_limits R (metric_topology R R_bounded_metric) seqx (apply_fun gR x) (apply_fun f x) HHaus HgxR HfxR Hneqxy Hseq_on Hnbhd_g Hnbhd_f).
L184658
Let x be given.
L184658
Assume HxX: x ∈ X.
L184658
We will prove apply_fun gR x ∈ I.
L184659
We prove the intermediate claim HmR: metric_on R R_bounded_metric.
L184661
An exact proof term for the current goal is R_bounded_metric_is_metric_on_early.
L184661
We prove the intermediate claim HTmR: topology_on R (metric_topology R R_bounded_metric).
L184663
An exact proof term for the current goal is (metric_topology_is_topology R R_bounded_metric HmR).
L184663
We prove the intermediate claim HI_sub_R: I ⊆ R.
L184665
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1).
L184665
We prove the intermediate claim Hm1R': (minus_SNo 1) ∈ R.
L184667
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L184667
We prove the intermediate claim HI_closed: closed_in R (metric_topology R R_bounded_metric) I.
L184669
rewrite the current goal using metric_topology_R_bounded_metric_eq_R_standard_topology_early (from left to right).
L184669
An exact proof term for the current goal is (closed_interval_closed_in_R_standard_topology (minus_SNo 1) 1 Hm1R' real_1).
L184670
We prove the intermediate claim HgR12: function_on gR X R ∧ uniform_limit_metric X R R_bounded_metric fn gR.
L184672
An exact proof term for the current goal is (andEL (function_on gR X R ∧ uniform_limit_metric X R R_bounded_metric fn gR) (continuous_map X Tx R R_standard_topology gR) HgRpack).
L184675
We prove the intermediate claim HgRfun: function_on gR X R.
L184677
An exact proof term for the current goal is (andEL (function_on gR X R) (uniform_limit_metric X R R_bounded_metric fn gR) HgR12).
L184680
We prove the intermediate claim HunifRgR: uniform_limit_metric X R R_bounded_metric fn gR.
L184682
An exact proof term for the current goal is (andER (function_on gR X R) (uniform_limit_metric X R R_bounded_metric fn gR) HgR12).
L184685
Set seqx to be the term graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x).
L184686
We prove the intermediate claim Hconvx: converges_to R (metric_topology R R_bounded_metric) seqx (apply_fun gR x).
L184688
An exact proof term for the current goal is (uniform_limit_metric_imp_converges_to_metric_topology_at_point X R R_bounded_metric fn gR x HmR Hfn1 HgRfun HxX HunifRgR).
L184690
We prove the intermediate claim HseqxI: ∀n : set, n ∈ ω → apply_fun seqx n ∈ I.
L184692
Let n be given.
L184692
Assume HnO: n ∈ ω.
L184692
We prove the intermediate claim Hseqxdef: seqx = graph ω (λn0 : set ⇒ apply_fun (apply_fun fn n0) x).
Use reflexivity.
L184694
rewrite the current goal using (apply_fun_of_graph_eq seqx ω (λn0 : set ⇒ apply_fun (apply_fun fn n0) x) n Hseqxdef HnO) (from left to right).
L184695
An exact proof term for the current goal is (HfnRange n HnO x HxX).
L184696
An exact proof term for the current goal is (converges_to_closed_in_contains_limit R (metric_topology R R_bounded_metric) I seqx (apply_fun gR x) HTmR HI_sub_R HI_closed Hconvx HseqxI).
L184703
An exact proof term for the current goal is Hseries.
∎